The Republican from South Carolina announced the resolution during a news conference.
"I have introduced a resolution today with [Senate Republican leader Mitch] McConnell and the purpose of the resolution is to let the House know that the process you're engaging in regarding the attempted impeachment of President [Donald] Trump is out of bounds, is inconsistent with due process as we know it ... and is a substantial deviation from what the House has done in the past regarding impeachment of other presidents," Graham said.
He said the impeachment investigations are "dangerous to the future of the presidency."
House Democrats plan to resume conducting closed-door depositions on Saturday, following a two-day break for Cummings' funeral.
The three panels leading the inquiry -- the House intelligence, oversight and foreign affairs committees -- were able to finish the deposition of Laura Cooper late Wednesday, hours after her testimony was disrupted by dozens of Republicans. She oversees Ukrainian and Russian policy at the Pentagon.
The GOP lawmakers staged a 5-hour demonstration at the Capitol, angered that the investigating Democrats are gathering testimony in closed-door sessions.
Rep. Mark Meadows, R-N.C., a member of the oversight committee, called Cooper a "credible witness" but said her testimony didn't offer any "groundbreaking revelations."
Several additional Trump administration officials are set to testify in the coming days -- including European and Eurasian affairs aide Philip Reeker on Saturday and former national security deputy Charles Kupperman on Monday.
Investigators are also scheduled to hear from Timothy Morrison, a National Security Council aide who was party to the July phone call between Trump and Ukrainian President Volodymyr Zelensky. The call was the subject of a whistle-blower report that spawned the impeachment review.
House Democrats heard Tuesday from Trump's ambassador to Ukraine, William Taylor -- who expressed concern the president may have attempted to leverage military aid to Ukraine for Kiev's help investigating 2020 presidential candidate Joe Biden and his son, Hunter, a former Ukrainian gas company executive.
Reports Wednesday said Taylor testified that Trump administration diplomat Gordon Sondland told Kiev the military aid depended on Ukraine investigating the Bidens -- a connection Trump, Sondland and White House chief of staff Mick Mulvaney have denied.