If the universe is expanding, it would make sense that the spaces between particles are getting bigger. If this is so, then the particles which make up atoms are also affected. Does that imply the spaces between the components of an atom will become large for the subatomic forces to hold? Are atoms getting weaker?
Any effect of the expansion of the universe on atoms is beyond all experimental hope. While spacetime should affect the movement of atoms (the metric is right there in all the equations), due to the constants involved, the effect is around the Planck scale, and the expansion itself is already pretty low in intensity. Hence any effect would require a long time before being observable. Just by being in the same room, you affect atoms around you more than the expansion of the universe.
Though that does not means this was always the case or always will be. The very early universe had a very strong gravitational field, and the Big Rip is a scenario where the expansion grows at such a rate that it will eventually affects all scales.
To understand the numbers, here are the four forces we know affecting all matter:
Look at the strenght of each interaction.
The quantum mechanical equations that define the nucleons/atoms/molecules/solid/liquid/gas phases incorporate in their potentials the strength shown in the column above. The strength of the gravitational force is so small that it will not affect the solutions within the possibility of measurement. The binding forces are very much stronger .Any diminution of the gravitational strength due to the expansion of the underlying fabric is unmeasurable because the expansion is even weaker than gravity, even galaxies retain their form:
When we go within galaxies they do not measurably expand, one has to compare the expansion rate with the strength of the gravitational interaction. Again expansion looses because in these huge masses gravitational attraction, even though very weak at the atomic scale, is much stronger than the expansion of the underlying space, within our possibilities of measurement. Expansion is seen in the motions of clusters of galaxies.
The expansion of the universe is about galaxies moving away from each other. There is no general increase of distance between stars within any one galaxy. There is nothing at all going on at even smaller scales, so far as cosmic expansion. That is the standard way of thinking in physics.
If atoms did change over time in a way related to cosmic expansion, we'd have trouble with defining units of time and distance. For the last couple decades, the second is defined certain vibrations of pure Cesium 133, and the fundamental space-time constant, speed of light, is a stated fixed number. If atoms did something odd like slow down slightly over millions of years - how would we know? So any effort to uncover novel new phenomena normally concentrate on pure numbers (without units) such as electron/proton mass ratio and the fine structure constant.
This gradual change is pure theoretical speculation. So far, unless I missed something, there's very little published experimental work on observing any such phenomena. (I didn't say 'none', but 'very little'...)
Everything in the universe tend to be more stable. Stability can either be attained by having maximum randomness or having less energy. Both the factors are independent in nature, the one with more influence predominates. In case of atoms, electrons in the atoms have less energy at lower orbits, the electrons in higher shell have more energy. Thus, electrons will be stable with it, without getting more randomness (entropy). Thus, you might not find increase in distance between the atomic components. So, atoms will not get weaker.