The Russian claims to the Arctic are preposterous in America's point of view. The Arctic is basically a giant ice shelf which is already severely compromised due to Global Warming causing a already noticeable rise in sea levels. Not to mention that the Arctic still remains one of the world most endangered natural ecosytems where many species are on the verge of extinction.
The Russians basically want to set up oil rigs and pollute the entire area selfishly destroying one of the world largest ice shelf causing a rapid increase in world wide sea levels and destroying an endangered natural ecosystem for profit! This is morally disgusting to civilized people, especially considering that Russia has no lack of oil resources in Siberia instead of damaging the whole world for a profit.
Another point is the Russian claim itself, which America does not accept. How can the Russians claim the right to drill there in the first place ? Proximity to the Arctic doesn't give Russia the right over anything just as Australia or Chile can't annex Antarctica just because they are close to it! If that were the case, the Moon should belong to USA as America is the first and only nation to send its men there. Will Russia accept USA annexing the moon ?
Exploration of natural resources of the Arctic can happen if there is an international consensus decided by the UN on how to divide the Arctic and what mineral rights Arctic circle countries have and what rights other countries have. It is no-man's land and that implies that no country can claim any rights over the Arctic unilaterally.