When Speaker Paul Ryan was asked if Greg Gianforte ought to withdraw from the Montana special House election, he responded with a rebuke of Gianforte's actions but stated that whether or not Gianforte makes it to Congress is the decision of Montana's voters.
Although Ryan has plenty of power as Speaker of the House, it seems he is right: Montana's voters, not the U.S. House, get to decide if Gianforte gets his seat.
Article I, Section 5 of the Constitution allows both houses of Congress to "be the judge of the elections, returns and qualifications of its own members," as well as to "determine the rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two thirds, expel a member."
Members have been denied their seat in the past for crimes. Socialist Representative Victor Berger was indicted on espionage charges in March 1918. Later that year, he won re-election, but the House refused to seat him.
Congress has also refused to seat members due to faulty election results. In 1974, Representative Louis C. Wyman appeared to have won a New Hampshire Senate seat. A recount awarded his opponent, John Durkin, the victory but Wyman appealed. After a second recount, Wyman was again awarded with a victory, but the Senate withheld his seat and called for a special election which Wyman lost.
But more recently, the courts have ruled that Article I, Section 5 in the Constitution does not give Congress indiscriminate power to unseat elected members.
Powell v. McCormack in 1969 limited Congress' ability under the membership clauses, holding that the two houses have "no power to exclude a member-elect who meets the Constitution's membership requirements," which are expressly prescribed in the Constitution -- living in the state, being at least 25 years old, and being a U.S. citizen. Body slamming a reporter doesn't change any of those facts about Gianforte. It also doesn't change the validity of the election results.
Congress does still hold broad authority to expel members who have already been seated. A document published by the Congressional Research Service explains that "while there are no specific grounds for an expulsion expressed in the Constitution, expulsion actions in both the House and the Senate have generally concerned cases of perceived disloyalty to the United States, or the conviction of a criminal statutory offense which involved abuse of one's official position."