I don't think you need such extreme relativistic speeds to obtain the desired result though. The above assertion that every proton would have the energy of a baseball puts us at a rather incredible speed. A quick Google gives the energy of a baseball pitched at 90 mph to be around 117.4 J. For a proton to have the equivalent relativistic kinetic energy would put it at 99.999999999999999999999918% the speed of light!!

Now let's take a much more reasonable speed like 10% the speed of light. Assuming the 100 protons per cubic meter figure above, each square metre of ship now only needs to dissipate 2.27 mW of energy. 10% the speed of light is enough time to reach Alpha Centauri within a single lifetime (42 years). And fast enough to visit every part of the galaxy in less than a million years. We could even imagine generational ships travelling at 1% the speed of light (now the energy dissipation demands are 2.25 μW per square metre of ship surface). That's still under 10 million years to colonise the entire galaxy.

If intelligent life is abundant in the galaxy then I don't think the speed of spaceships at least offers a fundamentally insurmountable technical challenge for that life to spread everywhere.