MOSCOW—President Vladimir Putin on Thursday defended Donald Trump from angry criticism following their summit in Helsinki, as the U.S. president called for a second face-to-face meeting with his Russian counterpart.

Mr. Putin, in his first comments in days, praised the Helsinki summit and said it led to some “useful promises.” Little more than an hour later, Mr. Trump tweeted that he wanted a follow-up meeting with Mr. Putin to work on the issues he said they discussed, including terrorism, Israeli security, nuclear proliferation...