WASHINGTON — Eric Shinseki’s resignation as head of the Department of Veterans Affairs was perhaps more welcome in the offices of worried congressional Democrats than in the suites of top Republicans who had pressed the scandal against President Obama.

“Today’s announcement really changes nothing,” Speaker John A. Boehner said on Friday shortly after Mr. Shinseki’s ouster, saying that congressional Republicans would continue to hold the president accountable for what transpires at the agency. “One personnel change cannot be used as an excuse to paper over a systemic problem.”

Mr. Boehner and Representative Eric Cantor of Virginia, the House’s No. 2 Republican, had refused to join the call for Mr. Shinseki’s head, since from their perspective, replacing one man at the top of a sprawling and long-struggling bureaucracy with another Democratic manager was not going to solve the problem.