I have a feeling that Ted Cruz will attempt to introduce impeachment charges against Obama over Benghazi and the immigration executive order once the new Congress is sworn in come January. I doubt that it would get that many votes however (possibly only from Jim Inhofe, Jeff Sessions, Mike Lee, Joni Ernst, Tim Scott and Ted Cruz himself).
Only the House can impeach.
My mistake, as I wasn't 100% clear on the impeachment procedure. Even though John Boehner probably won't bring up impeachment charges for a vote in the House, the charges would probably pass easily. Once the impeachment process moves onto the Senate, the only people who would vote to convict Obama would be Ted Cruz and the other 5 Senators that I listed in my earlier post.
