Q:

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:

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.

