The implication is that Russia simply doesn't have anything on the grid that's capable of boosting a rocket to Mars. Given the extensive cooperation between the US and Russia, it can be assumed that NASA would be well aware of anything they have that's suitable and that NASA considered it and rejected it (offscreen) along with ruling out the capabilities of the other space-capable organisations such as the European Space Agency, Korea Aerospace Research Institute, Japanese Space Agency and Indian Space Research Organization, none of whom get more than a passing mention in the story.
Russia is also, presumably, party to the treaties discussed by Guo Ming:
GUO MING, director of the China National Space Administration,
examined the daunting pile of paperwork at his desk. In the old days,
when China wanted to launch a rocket, they just launched it. Now they
were compelled by international agreements to warn other nations
It was a requirement, Guo Ming noted to himself, that did not apply to
the United States. To be fair, the Americans publicly announced their
launch schedules well in advance, so it amounted to the same thing. - The Martian
Note that it's actually a complete coincidence that the Chinese have anything that can make the interplanetary journey:
Guo Ming stood and pinched his chin. Pacing, he said, “We can really
send the Taiyang Shen to Mars?”
“No, sir,” said Zhu Tao. “It’s far too heavy. The massive heat
shielding makes it the heaviest unmanned probe we’ve ever built.
That’s why the booster had to be so powerful. But a lighter payload
could be sent all the way to Mars.”
In this interview with nautil.us, Weir makes it clear that he doesn't think there would be a philosophical objection to the Russians helping NASA:
You say you’re a pessimist, but your book is a pretty optimistic one.
On the grand scheme of things with dystopia at zero and utopia at 10, I’d say I predict about a six or a seven. The thing to remember is The
Martian doesn’t take place in some far-off, distant, unimaginable
future. It takes place maybe 20 years from now. It’s not like, oh, I
just took my flying car to work today. Yes, there’s cooperation with
the Chinese, but I think we should be cooperating with them more on
space stuff, and I think it’s inevitable. Bear in mind, at the height
of the Cold War we cooperated with the Russians on space stuff. It’s
just a thing we do! I have more faith in humanity maybe than others
do, and there is a general optimistic kind of feel to the book, like
we can do this, and people tend to inherently want to work together
when there’s a problem.