Is there a natural process by which hydrogen is generated from heavier elements in the cosmos? we know that stars fuse hydrogen into helium starting at 3 MK; 13 MK in the Sun's core; 
carbon fusion starts at above 500 million K, and silicon fusion starts at over 2700 million K for comparison;
we know fusion stops at iron, because a star has to use more energy to fuse that than it gets back;
so heavier elements are created mostly in a supernova (but also possible in small quantities by special processes like neutron capture);
finally sun-like stars end up as white dwarfs, bigger stars as neutron stars, quark stars, black holes;
and black holes ultimately convert themselves into radiation, in the far distant future when the stable black hole mass limit goes up high enough that even the most massive black holes evaporate;

http://en.wikipedia.org/wiki/Vacuum_Diagrams

so my question is, will it be like Stephen Baxter said, that in the future only radiation will be left in the universe?
Specifically, is there a natural process out there by which hydrogen is spewed into the cosmos, converted back from heavier elements, to regenerate the fuel for stars so that they may shine in the far distant future as well?

Of course we don't need to worry about this for the time being. This is only considering our concern with what will be 10^70 years from now.
There are a couple of relevant questions one would want to ask:

1) Do protons decay, and if so, what do they decay into? The answer appears to be no, or at least the theoretical lifetime of the proton must increase as a results of these experiments. If they do, eventually the universe could end up in a state of radiation (and dark energy, and dark matter, unless they also decay). 

2) Is Hydrogen a bi-product of any natural decay process? Below is a table of all known nuclides.



As you can see, the majority of the elements (not necessarily by number or mass in the universe) do decay through some type of process. There exists a 'stable' ridge (called the island of stability, surrounded by the sea of instability) of elements which will happily exist forever.

The question is, which modes of decay produce protons (Hydrogen nuclei)? Well, there is proton decay (not the proton itself decaying), which is colored in red, though I have to admit that I don't know exactly what this refers to. Fission bi-products are gamma rays (high energy photons), neutrons, and daughter nuclei (see Decay chain). Though, I should mention that free neutrons produced from this type of radioactive decay are not long lived, decaying into a proton and an electron (this process takes on average approximately 11 minutes). By this logic, also isotopes which decay by emitting neutrons, colored purple, would also eventually produce protons. $\beta^{-}$ and $\beta^{+}$ refer to the beta decay process, where the minus sign refers to the emission of an electron and the plus sign refers to the emission of a positron (the anti-particle of the electron). $\alpha$ decay is the emission of a Helium nucleus, which is stable.

Now, given that there are ways for heavy elements to naturally produce protons, the question I would ask is what is the rate of these processes in the universe compared to fusion processes occurring at the centers of stars. I'm not sure that I could give you an answer to this question (or even point you to the appropriate material), but in principle these rates are known. I'd imagine that it'd be quite a lot of bookkeeping to get it correct.