WASHINGTON—Top congressional Republicans sharply criticized President Donald Trump Monday after he cast doubt on Russian interference in the 2016 election during a press conference with Russian President Vladimir Putin.

“Missed opportunity by President Trump to firmly hold Russia accountable for 2016 meddling and deliver a strong warning regarding future elections,” Sen. Lindsey Graham (R., S.C.), who often advocates for Mr. Trump on Capitol Hill, said on Twitter Monday. “This answer by President Trump will be seen by Russia...