“Is it fair for the president of the United States to give American businesses an exemption from his health care law’s mandates, without giving the same exemption to the rest of America? Hell no, it’s not fair,” Speaker John A. Boehner told a closed-door gathering of House Republicans on Tuesday, according to those present.

Some Democrats were also dismayed by the White House’s actions. Senator Tom Harkin of Iowa, the chairman of the Senate Health, Education, Labor and Pensions Committee and an author of the health law, questioned whether Mr. Obama had the authority to unilaterally delay the employer mandate.

“This was the law. How can they change the law?” he asked.