President Trump's decision to congratulate Russian President Vladimir Putin — after he was advised not to, as CBS News has reported — has elicited responses ranging from incredulity to indignation from members of his own party.

On Tuesday, Mr. Trump volunteered to reporters that he had congratulated Putin on his victory, a gesture that has drawn criticism for multiple reasons. There are widespread accusations of ballot stuffing in the election that lead to Putin's win. The U.S. has also said it stands with the United Kingdom, after the U.S., U.K., Germany and France have said they suspect Russia is behind a nerve-agent attack in the U.K. Adding to that, only moments after Mr. Trump announced he had congratulated Putin, both Republicans and Democrats on the Senate Intelligence Committee held a news conference claiming Russia unequivocally attempted to intervene in the 2016 election. At that press conference, no Republican — or Democrat — ventured to comment on Mr. Trump's call with Putin when asked.

Sen. John McCain, R-Arizona, blasted out a statement criticizing Mr. Trump's words as an insult to all Russians who were deprived of a free and fair election. Later Tuesday, White House press secretary Sarah Sanders would not say whether the White House believes Russia's election was free and fair: "We don't get to dictate how other countries operate," she told reporters.

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"An American president does not lead the Free World by congratulating dictators on winning sham elections," McCain said in a statement. "And by doing so with Vladimir Putin, President Trump insulted every Russian citizen who was denied the right to vote in a free and fair election to determine their country's future, including the countless Russian patriots who have risked so much to protest and resist Putin's regime."

Sen. James Lankford, R-Oklahoma, told "CBS This Morning" he was "caught off guard" by the president's words.

Sen. Lindsey Graham, R-South Carolina, speaking with reporters on Capitol Hill, called Russia's election a "sham election," but said he recognizes Mr. Trump has to work with him. Graham did say it was "disappointing" Mr. Trump didn't mention the nerve attack.

"It's clear to me we are not sending the right message to Putin about his behavior in our own backyard and we had a chance to reinforce an ally and we missed that opportunity," Graham said.

Sen. Chuck Grassley, R-Iowa, said he wouldn't comment on Mr. Trump's call with Putin — but then said he "wouldn't have a conversation with a criminal," pointing to Russia's alleged actions.

"I think Putin is criminal," Grassley told reporters. "What he did in Georgia, what he did in Ukraine, what he's done in the Baltic. What he's done in London poisoning people with nerve gas that's a criminal activity."

"I wouldn't have a conversation with a criminal," he said.

Florida Sen. Marco Rubio didn't think much of the president's call but objected more to the reports that he was advised not to congratulate Putin.

"You know what I like even less? There's somebody close to him leaking this stuff out. If you don't like the guy quit. To be this duplicitous and leak stuff out, it's dangerous. So I don't like what he did," Rubio said. "But I really hate that's there's someone in his inner circle that's willing to leak this stuff. If you don't like working for the president you should resign your job."

Many senators and House members simply haven't commented.

CBS News' Alan He contributed to this report.