Do you think the United States still participates in the "space race"?
The "Space Race" as originally defined exists no longer however the U.S. is committed to a long term space exploration.
The original space race was first to get a man in orbit which the Russian's did with Uri Gagarin, and then being the first country to reach the moon which of course the U.S. did. Since then the space mission of both countries has be redirected.
Even before the fall of the USSR the United States and Russia were engaging in joint space missions as both sides realized their desires in space were basically the same. Both sides wanted a series of communications satellites, weather satellites, and geopositioning satellites.
And even though each side has its share of military satellites in space neither seems to be trying to one up the other as was happening in the 1960s and 1970s. One simple reason is the economics of doing so is prohibitive.