WASHINGTON — The Republican chairman of the House Intelligence Committee was strongly criticized on Thursday by the panel’s ranking Democrat for refusing to give up his subpoena power over the investigation into Russian interference in the 2016 election less than two months after announcing he would step away from the inquiry.

The chairman, Representative Devin Nunes of California, on Wednesday also issued three subpoenas to law enforcement and intelligence agencies seeking information about the so-called unmasking of associates of President Trump whose identities were incidentally caught up in surveillance of foreign operatives. The subpoenas resurfaced Mr. Trump’s dubious claims about Obama-era surveillance at a time when Mr. Trump is facing serious questions about whether he tried to interfere in the F.B.I.’s investigation.

“When someone says they’re going to recuse themselves or step aside from the investigation, you have to expect that they’re not going to insist on having final approval over subpoenas, one of the most important tools of an investigation,” Representative Adam B. Schiff of California, the committee’s top Democrat, said in an interview.

Mr. Nunes’s issuance of subpoenas was followed on Thursday morning by a Twitter post from Mr. Trump on the same subject. “The big story is the ‘unmasking and surveillance’ of people that took place during the Obama Administration,” he said.