“Sickening to learn this morning of another mass murder in Dayton, Ohio overnight,” he tweeted on Sunday. “Two horrifying acts of violence in less than 24 hours. We stand with law enforcement as they continue working to keep Americans safe and bring justice.”

But Senate Democrats may not get off easy by just blaming Mr. McConnell. The spate of mass shootings in recent years has led an array of activist groups to adopt positions that were once relegated to the far fringes of the gun debate; these groups now aim to hold accountable Democrats who they believe do not fight hard enough for gun control measures.

[A weekend of gun violence stuns the country]

“They must do everything in their power, everything in their capacity as a senator, from holding a filibuster, to placing a hold on nominees and key bills, to getting back to D.C. right now to show the nation that they are willing to act,” said Igor Volsky, the executive director of Guns Down America. “It’s not enough to simply ask Mitch McConnell to bring something to a vote. They have to use their leverage as a United States senator to shame him into taking action.”

Senator Chris Murphy of Connecticut, who has pushed for gun control legislation since the 2012 elementary school massacre in Newtown, Conn., said Sunday that he’d spent the weekend discussing gun control proposals with his Republican colleagues.

If Republicans won’t budge, then Democrats will “have to get together and figure out what leverage we have in the Senate,” Mr. Murphy said in an interview. But realistically, he said, gun control may only pass the Senate if Democrats control the chamber.

“It’s my responsibility to try to find a legislative path, but I’m not going to take too many kicks at the football if it keeps getting pulled out from under us,” Mr. Murphy said. “I think we have had our eyes firmly on the electoral path since the spring of 2013, and we probably have to keep it there.”