This will never happen.

Rand Paul has gone on record as saying that Republicans should not investigate other republicans.

Mitch McConnell said on a CNN interview after Trump's speech to Congress that "The people elected Trump so he could sign off on the Republican Agenda", or something along that line. Basically, he was admitting that the Republicans consider him nothing more than someone there to rubber stamp whatever they want to push through.

The very first thing that the Republicans tried to do when they came back was to dismantle an ethics/oversight committee. That alone is a pretty strong indication that they view the next four years as a GOP-Free-For-All, and have no intentions of letting anything get in the way of them running roughshod all over the place.

Further, look how many high-ranking Republicans have already been exposed as having ties to the various Russia scandals. It's very likely that a full, fair, and independent investigation would reveal that the ties to Russia and the associated corruption go much deeper than any of us are aware, which would likely lead to several Republicans being forced out of office or outright jailed. There is absolutely no way they are going to allow that to happen, and unfortunately, they are in a position where they can ensure that a full investigation will go absolutely nowhere.

Republicans are already on the hot-seat over Obamacare, as evidenced by what has been happening at the various GOP "town halls" recently. The Republican voter base is beginning to realize that "fixing Obamacare" isn't as easy as they thought it was, and will likely lead to them losing some or all of their healthcare, or at least having to pay a lot more for what they've got. Between that and the whole Russia thing, I believe that if the GOP were to allow a full, fair, and impartial investigation to happen with no obstruction from the GOP, you'd not only see Trump removed from office, but you'd also see a whole bunch of Republicans lose their seats in 2018.

Republicans will never let that happen. Ever. They have become very good at being obstructionists, and will apply what they've learned to ensure that no real investigation ever takes place in order to protect their own necks.