The Republican senator Lindsey Graham, a top ally of Donald Trump, introduced a resolution condemning the Democratic-controlled House for pursuing a “closed door, illegitimate impeachment inquiry” as evidence against the president continues to mount.

Graham announced the move on Thursday afternoon and said the Senate’s top Republican, Mitch McConnell, is a co-sponsor of the resolution, demanding that Republicans be given more chances to question witnesses.

The nonbinding resolution gives Senate Republicans the chance to show support for Trump at a moment when the president is urging his GOP allies to get tougher and fight harder for him as the House impeachment inquiry gathers momentum a month into the process.

“If you can drive down a president’s poll numbers by having proceedings where you selectively leak information, where the president who’s the subject of all this is pretty much shut out, God help future presidents,” Graham said.

Rather than attacking the substance of the evidence investigators are accumulating, Republicans have focused on asserting that House impeachment proceedings have been secretive and unfair.

Graham said: “I’m not here to tell you that Donald Trump’s done nothing wrong.”

Trump tweeted his thanks for House Republicans, who “stormed” a closed-door committee hearing on Capitol Hill on Wednesday, disrupting a crucial deposition related to the Ukraine controversy.

The chaos and confusion temporarily shut down the proceedings before the three House committees leading the impeachment inquiry as Republicans tweeted updates of the disruption from their cellphones, which are not permitted in classified areas. Their presence in the chamber reportedly erupted into yelling matches with committee members.

“Thank you to House Republicans for being tough, smart, and understanding in detail the greatest Witch Hunt in American History. It has been going on since long before I even got Elected (the Insurance Policy!). A total Scam!” Trump tweeted.

Many Senate Republicans are taking a wait-and-see approach amid the House impeachment inquiry. Rather than commenting on the substance of the inquiry, they are united in complaining that the House is taking depositions from witnesses behind closed doors.

The escalation by Republicans comes after Bill Taylor, the most senior US diplomat in Kyiv, testified for hours before House investigators on Tuesday, delivering an account that was so shocking to some lawmakers that freshman Democrat Andy Levin described it as “my most disturbing day in Congress so far”.

In a lengthy opening statement, Taylor said Trump wanted “everything”, including military aid to Ukraine, tied to a commitment by Ukrainian leaders to investigate Democrats and the 2016 election plus a company linked to the family of Trump’s leading 2020 Democratic rival, Joe Biden.

Taylor said Trump “wanted President Zelenskiy ‘in a public box’ by making a public statement about ordering such investigations”.

Republicans have largely sought to attack the inquiry on procedural grounds, objecting to the private nature of the hearings and demanding access to the full breadth of the testimony that has rattled Washington in recent weeks.

Much of the testimony that has been made public, however, and news reports confirm key elements of a whistleblower complaint that set in motion the impeachment inquiry. The investigation centers on reports of Trump withholding military aid and dangling a meeting at the White House for Ukraine’s president, Volodymyr Zelenskiy, in return for favors that would benefit him in domestic US politics.