Is the fine-structure constant related to the size of the observable universe? The fine-structure constant $\alpha \approx 1/137$. In Planck units, this is also the charge of the electron squared, $e^2 = \alpha$ ($e \approx 0.085$). In Planck units the size of the observable universe is about $10^{60}$ along the spatial and time axes. My question is, is the logarithm of the size of the observable universe, $\log(10^{60}) \approx 138$ related to the fine-structure constant (electron charge)? I suppose there is some relation since the observable mass is observed by measuring electromagnetic radiation, which is generated by charged particles like the electron.
 A: It would be difficult even to construct a theory in which those quantities are related in that way, for a number of reasons. 
The immediate problems have been pointed out in other responses: ~1/137 is just the infrared (zero energy) value of the fine structure constant, the high energy value is usually regarded as fundamental (@knzhou); the current size and age of the universe is just a temporary quantity, and the fine structure constant does not exhibit the time dependency one would expect if there was such a relationship (@knzhou); and - in my opinion the severest difficulty - the proposed relationship depends on measuring the universe's size and age in Planck units, whereas one normally looks for fundamental quantities to be dimensionless, e.g. a ratio of sizes or a ratio of ages, since such ratios remain the same regardless of the system of units (@Dale). 
For these reasons, if one really were trying to motivate the value of 1/α as being approximately ln 10^60, it might make more sense to obtain "~10^60" as 1/sqrt(10^-122), where 10^-122 is a dimensionless parameter characterizing the cosmological constant. 
Nonetheless, if one were determined to try to construct a theory in which the OP's relationship was not a coincidence - a few pointers. 
Counting time in Planck units
Supposing that the exact age of the universe in units of Planck time could be of fundamental significance, is a speculation one would associate more with attempts to create a "digital physics" e.g. universe as cellular automaton, than with the role that Planck units actually play in mainstream physics. There, Planck units have more to do with reasoning about "naturalness", by telling you what the expected size of various properties of the universe would be, if all the fundamental constants were of order 1 in size. But this is only used for order of magnitude estimates. 
However, I can think of two theoretical frameworks in which Planck-time time-steps are actually supposed to play a role. There are a number of papers by Paola Zizzi in which there's a kind of quantum-circuit model of inflation; and then in 2017 a group including Sean Carroll came out with a similar-looking story. 
Meanwhile, there is at least a precedent for the idea that "time variation of the fine structure constant [could be] driven by quintessence", quintessence being a hypothetical scalar field responsible for dark energy. So all you need is a quantum-information model of holographic quintessence, in which UV/IR mixing allows quantum gravity to play a role in determining the infrared value of the electromagnetic coupling constant... 
Current size and age of the universe
However, even if you could construct such a theory, you would run up against the evidence that the fine structure constant simply hasn't been varying with time. So you could instead try to revive a static or a steady-state cosmology, in which "10^60    Planck units" is somehow a fixed property of the universe, rather than just how big    and how old it happens to be right now. I am aware that there are specific proposals that are still out there - we live in an eternal rotating "Godel universe", the CMB is the thermal equilibrium of some ubiquitous astrophysical source rather than a fading big-bang remnant - but these ideas must be heavily at odds with modern cosmological data and paradigms, if I am to judge by their fringe status. 
There is some mainstream work on "cosmic coincidences" (numerology involving cosmological parameters) in which the current age of the universe does appear, e.g. this paper from 2000. This is called the "why now" question. However, again,  this only involves orders of magnitude - the argument is considered a success if it can give a reason why the coincidence occurs after 10 billion years, rather than 1 million years or 1 trillion years.
There have been questions asked on this site, regarding relations between the cosmological constant, and the current age and current size of the universe. Conceivably they lead to theoretical avenues whereby a relationship between fine structure constant and cosmological constant could mimic the proposed relationship between fine structure constant and size/age of the universe.  
Conclusion
I think that concludes my attempt to make this work. I never even got around to contemplating exactly how quantum electrodynamics might be modified in order to introduce a dependency on any of these cosmological parameters, beyond a handwave in the direction of quintessence. Quantum field theory contains many ways in which one quantity may be made dependent on another; but the basic problem here is that it is hard to see how these particular quantities (current age/size of the universe, measured in Planck units) can enter into that interplay at all. It would seem to require some quantum-gravity magic and perhaps even an abandonment of big-bang cosmology. Hopefully I have at least conveyed something of how a real theorist might proceed, if they put aside the professional knowledge which tells them this idea leads nowhere, and spent five minutes trying to make it work. 
Incidentally, the answer by Sean Lake touches on something much more real, namely, the likely existence of order-of-magnitude relationships between the size of the fine structure constant and the size of the universe in the era with galaxies, atoms, and life. It would be good to have someone who really knows this material, spell that out quantitatively. 
A: What you found was a coincidence. We know this because the fine structure constant does not have units, so everyone agrees on its value, but the size of the observable universe does, and you can't get from one to the other just by taking a logarithm.
There is a way in which the size of the observable universe is related to the fine structure constant, but only if you define "observable" as being "directly observable with light only" (gravity waves and neutrinos can go further).
In the case where we mean "observable using light only", the size of the observable universe is set by when the cosmic microwave background radiation began to move freely because the plasma that filled the universe cooled enough for the hydrogen and helium to become neutral. For reasons unknown to me, we call that event "recombination" (unknown because the hydrogen and helium atoms were never combined before, so the "re-" seems inappropriate). I do not know the details, myself, but I am confident that the fine structure constant plays an important role in determining when that happens (alongside the mass of the electron and proton). So, if $\alpha$ were smaller, recombination takes longer but the universe can also become transparent sooner, while if it's higher recombination happens sooner but it takes longer for the universe to become transparent (if ever).
I add the "if ever" because the equivalent of $\alpha$ for the strong force is somewhere around $1$ and the universe has never been transparent to gluons - we went straight from quark gluon plasma to confined gluons that cannot move through freely through space without instantly producing a bunch of quarks, as far as I know.
A: As a historical addendum to the other answers, the question is reminiscent of the "large number coincidences" that intrigued some physicists in the early 20th century and led to the cosmology of Dirac's large number hypothesis. They had observed that when you construct dimensionsless quantities out of physical constants (being dimensionless their values are independent of your measurement system) they either take on values $\approx O(1)$ or very large values $\approx O(10^{40})$ (or their reciprocals). This was especially intriguing since some of the large values seemed to be on the order of magnitude of the size of the observable universe or the number of particles in it. Dirac suggested that some of these values were equal, producing a cosmology with changing gravitational constant and increasing mass over time since he knew the universe was expanding. This model never truly panned out, and we know from observations of remote stars and the Oklo reactor that the constants seem constant over multibillion year timescales.
This is dealt with extensively in the excellent The Anthropic Cosmological Principle by Barrow & Tipler. They also point out that some of these coincidences are mildly unsurprising given our own existence (anthropic fine-tuning). Observers likely cannot form in very small universes, so we need to have very weak gravity in order to have large and slowly changing structures on which faster processes (due to electromagnetism and nuclear forces) can occur and produce observers. The ratio of the fine structure constant to the strength of gravity is $\approx 10^{36}$, which gives ample time for such emergence. See also (Carr & Rees 1979) for a good take on this. 
A: 
My question is, is the logarithm of the size of the observable universe, log(10^60)≈138 related to the fine-structure constant (electron charge)? 

You could pick units such that the log of the size of the universe in those units was exactly $1/\alpha$. Or you could pick units so that it was equal to $\alpha$ directly. Or you could pick units so that it was some power of $\alpha$ or you could pick units so that it was the log in some different base. Or ...
In all of those units it would be “related”. The relationship is pure numerology, with no meaning independent of our arbitrary choice of units. 
