Is relativity necessary for the existence of life? If the universe didn't have the relativity principle, would it be able to support life?
Life consists of very complicated organisms. The operation of these organisms depends on the laws of physics.
If the laws of physics depended on absolute velocity, then it seems that life would have a more difficult task; organisms would have to adapt their biochemistry to the different absolute speeds of the planet as it moves with or against the motion of the sun around the galaxy.
If the laws of physics depended on the absolute gravitational potential, or on acceleration, then the biochemistry of life would have to adapt to the different accelerations / gravitational potential as life colonized higher altitudes. In addition, there would be a seasonal effect as the earth moves closer and farther away from the sun.

I think there's a way this question could be answered quantitatively. Begin with a modification of general relativity such as the post Newtonian parameters. See wikipedia article:
http://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism
Now analyze an important biological molecule whose shape is extremely important to life such as hemoglobin. Find out what range of post-Newtonian parameters are compatible with the operation of that molecule.
Unfortunately, I suspect that this is a research problem. If someone solves it, I presume they will publish it.
 A: one could tackle your question a little more formal if one looks at the limit $$c\rightarrow\infty\, .$$
We can now ask what happens in the several theories and what implications there would be for life.
A naive attempt
One starting point is indeed the Wikipedia article on physical constants. You can look for your favourite one depending on the speed of light and look on the corresponding site what it implies.
Ok, lets do this for one thing I just have chosen (almost) randomly: The Rydberg constant $R_\infty$.
It is given by $$R_\infty = \frac{m_e e^4}{8\epsilon_0^2h^3 \mathbf{c}} \approx 1.097 \times 10^7 m^{-1}\, .$$
As the article states, it is the most accurately measured fundamental physical constant and its value can be derived from first principles. Interesting to know, I always thought this was the fine-structure constant $\alpha$.
We state that $$\lim_{c\rightarrow \infty}R_\infty = 0\, .$$
The Rydberg constant has its interpretation as the lowest wavenumber $\lambda_{ion} = 1/R_\infty \rightarrow \infty$ that can ionize the hydrogen atom. This is linked to some lowest energy $$E_{ion} = c \frac{h}{\lambda_{ion}} \rightarrow  \frac{m_e e^4}{8\epsilon_0^2h^2}\, .$$
So it seems like we have won nothing at all. The wavelength goes to infinity but the corresponding energy remains constant. Or do we run into further problems because we have to look at the permittivity of vacuum $\epsilon_0$ that is also linked to $c$ via $\mu_0\epsilon_0 = 1/c^2$. This is puzzling - we cannot answer the question from this viewpoint, but earned a nice hint due to the fact that all we are discussing about corresponds to  wave propagation in electrodynamics.
Electromagnetic waves
Wave propagation at a certain frequency $\omega$ through vacuum is described by the Helmholtz equation
$$\Delta A_{\mu} + \frac{\omega^2}{c^2} A_{\mu} = 0$$
which also holds for quantum electrodynamics as discussed in another thread. Here we can see clearly what happens if $c\rightarrow \infty$:
The Helmholtz equation reduces to the Laplace equation
$$\Delta A_\mu = 0$$
which can be interpreted in a way that everything that happens, does it instantaniously - there is no retardation anymore. This implies that everything happens at the same time, at least in electrodynamics. In fact, this should also hold for all (effective) theories that can be described by interactions via light particles, or other massless particles since they also travel at $c$.
So to speak, the speed of light is something like a translation between space and time. If $c$ goes to infinity, maybe even the notion of time (and energy as the associated quantity) does not make sense.
I don't know what will happen, but one of the two cases seem to be plausible if $c\rightarrow\infty$:
Either all will happen instantaniously, or, maybe worse, everything will have to remain in a unchanged forever (static) - I don't think that life as we think of it is possible under these circumstances.
Sincerely
Robert
A: This question has been modified into a more specific form concerning two structures: Haemoglobin and the PPN Formalism for Gravitation. So I shall make some general comments about how to consider this: a linked Supplementary question might then be the best means to progress.
Haemoglobin is a large (class of) molecules, which have high complexity. Approx molecular weight 68000 (where Hydrogen=1). They are in many terrestrial life forms and they are a key component of blood. There are undoubtedly many outstanding scientific questions concerning the origin and dynamics of such a large molecule. One aspect in particular is the whole issue of "protein folding" which would help form them efficiently and successfully.
In terms of physics there are two aspects: Fundamental physics aspects and local environment aspects. The latter are the most important in practice with e.g Temperature undoubtedly being important; although pressure (ie blood pressure) is likely a function of local gravitation too. So there is the whole question of the biomechanics of a body in differing gravitational fields (surface of Earth, surface of Moon, in Vacuum, on Jupiter, etc) to consider. As the body is the "manufacturing unit" for this molecule its well being is important too - and Vacuum conditions and Jupiter conditions are generally considered hostile to life as we understand it.
In terms of the fundamental physics and laws, the most important here are those of Chemistry which come from Quantum Mechanics. In a hypothetical different universe with a different electric charge for example undoubtedly different molecules would form; perhaps no molecules can form, only atoms.
In terms of our universe (whatever gravitational laws really apply: Newton, Einstein, etc) the local strength of the gravitational field will be an important parameter in the body formation. In terms of biochemistry it probably only has an indirect effect from its contribution to environment (air,sea) pressure and other thermodynamic variables.
All these issues from the persective of life history and presence on other planets and environments is important research in the field of astrobiology.
The PPN formalism was invented to compare Einstein's General Relativity with competing theories via delicate astronomical measurements: but GR remains the best theory in terms of these tests. The best way for a gravity theory parameter to affect quantum parameters would be if, in a theory of Quantum Gravity, it was found that electric charge,say, was a function of the Gravitation constant (which then might not be constant). So that theory might have additional solutions with unusual parameters which might or might not allow the formation of complex molecules.
A: First, the principle of relativity is the statement that the laws of physics are invariant in all frames of reference.
So now we have physical laws that are a function of the frame of reference, the answer to your question would ofcourse depend on exactly what laws you replace them with. If there were only some minute change in physical constants throughout space, we would not necessarily have a problem, I am proposing we wouldnt even have a theoretical problem, as many of these numbers seem arbitrary in the SM.
Infact there are many more radical changes you can do to the physical laws that would not result in any (immediately) observable change at all, just think of string theory or most TOE ;)
A: The main problem is that the sun wouldn't work - no mass-energy conversion. Recall all the unusual and baroque theories of the source of solar radiation before relativity and QM (a giant coal furnace, gravitational collapse, ...).
A: There would be the problem of $E~=~mc^2$ and solar power.  We might imagine though some non-relativistic form of nuclear energy.  There is also the likely result that the spectra of elementary particles is determined by gravity.  QCD on the boundary of an AdS spacetime is equivalent to the symmetries of the graviton in the interior.  So without relativity the world might appear very differently.
This question is a bit of a “what if?” sort of question.
A: Special Relativistic dynamics is the reason that electrons have spin and behave as fermions. These properties of the electrons are the reason the various elements have different chemical properties or in other word the reason we have chemistry. Chemistry is the reason we have molecules. Without molecules there would be no life, thus without Special Relativity there would be no life.   
A: another possibility is that, assuming life could evolve into highly intelligent and technically capable lifeforms, it could then propagate at exponentially fast rates and take over the universe.
A: Even with relativity organisms will meet different gravitational potentials at different altitudes. This reduces air density and pressure and causes breathing difficulties for nonadapted species.
Without General Relativity (or a similar theory) we might also have a problem putting together a Cosmological Theory. Thus the Universe might not get started at all....
