Basically, not much.
I agree very much with what Vitalii Dem’ianiuk said here before. Ukraine is moving toward independence from Russia simply because the relationships are so bad. Economically, you can say this doesn't make much sense. Russia has plenty of gas, and it can produce gas cheaply if we looked upon it as an ordinary enterprise, but it doesn't work. There's no good relationship between these two. As I mentioned, it's quite obvious that Russia wants to avoid taking any gas through Ukraine in the future. Russian sales to Europe are falling. Russia is simply mishandling its market in Europe very badly, not being the least oriented towards it. The positive side of it is that Gazprom therefore cannot cause many problems, but there are specific issues.
Some of the biggest gas storage in the world is in western Ukraine; 36 billion cubic metres of gas can be stored there. Gazprom has an enormous interest in seizing these big storages for its supplies in eastern Europe if they were to be serious about it. We could see a sudden change of Gazprom coming in. Ever since Ukraine became independent, there has been a discussion about the status of its gas transit system through Ukraine so that it be owned by Russia or Ukraine. The current legal rule is that it must be Ukrainian state-owned, but there has been constant discussion about a consortium with Germany, Ukraine, and Russia, or with the European Union, Russia, and Ukraine. Because of the complete lack of trust between Russia and Ukraine, nothing has come out of these many discussions.
The short answer is that Gazprom cannot block it right now, but you never know how it can pop up.