WASHINGTON — Republicans scrambled on Tuesday to mitigate the damage done by President Trump’s embrace of President Vladimir V. Putin of Russia over his own intelligence agencies, setting a public hearing in the Senate next week, examining new sanctions on Moscow and reaffirming the fraying Western alliance.

“I think it’s important for the Europeans to know how the Senate Republican majority feels about the structures that were painstakingly set up over decades that have worked to maintain world peace,” said Senator Mitch McConnell of Kentucky, the Republican leader. “So I just think it was important for our friends and allies to hear from us.”

After Republicans pleaded privately with the White House for Mr. Trump to clean up his remarks, the president walked back his comments, asserting that he had misspoken a day earlier in Helsinki, Finland. But amid a flurry of overlapping policy proposals and a drumbeat of Democratic demands for swift action, the ability of congressional Republicans to move beyond statements of condemnation to legislation and oversight that could change Mr. Trump’s course was very much in doubt.

Many Republicans found themselves wrestling with an unwelcome dilemma two years in the making: They could publicly undermine the president and risk upsetting the loyal Trump voters they need to win elections, or they could stifle their own long-held beliefs that Russia is a dangerous actor set on destabilizing the United States. At least some were dealing with a moment they helped create: Since the summer of 2016, they have abetted the president’s refusal to recognize Russia’s actions by initiating their own attacks on those who would expose the Kremlin’s efforts.