One one hand, it seems like all major ideological disagreements have been solved, so there shouldn't be any reason for conflict, on the other hand, as we all know, countries with similar ideologies are able to compete, hate, and even fight directly against one another as well; and they did so many times throughout history. Russia and USA are still able to hate one another, as we can see from recent developments, plus, if you remember the 1948 Dawes Plan called not only for elimination of communism, but also for minimization of Moscow's influence as a separate objective.
If RF and USA decide to seriously compete against one another, the mass media on both sides will find plenty of arguments for hate and the ordinary people will follow. The real incentive for competition will be simple imperialist world suprmeacy, and nothing else really.
Also it has to be noted that one of the side-effects of globalization is peace and friendly relations. Both Russia and China might have become simply too entrenched in the world market and transnational corporate agreements. Cutting links with foreign markets will hurt both East and West a lot, hence it is questionable whether either side would risk worsening the realtions to that point. Just imagine what would happen if China and US started embargoing each other.
On the other hand, it can be done. If Russian economy becomes strong enough (about the size of India at least), it might then become possible for RF and PRC to start limiting their market activities in pro-western countries and focusing only on the markets in the "Axis of Evil" bloc and neutral countries like India. It will still hurt, no doubt, but the West will be hurt as well.
So, if (and only if) Russian economy really starts catching up to the level of the developed western European countries, it seems possible to me that a new Cold War could start with RF and PRC on one side, while US, EU and Japan on the other. Indian politics will also be crucial, but it is unclear so far which side it would join if such conflict was to occur. Russia would then join Axis of Evil's efforts and together with China start funding the rebels in Iraq and Afghanistan with arms and modern technologies, and then the US position in the Middle East will be seriously screwed.
How possible is this?