WASHINGTON—The demise of immigration legislation for the year and a Supreme Court decision creating a religious exception to the health-care law are the latest setbacks casting clouds over President Barack Obama's agenda.

The developments leave the administration with limited options. So far, the White House response has been to continue to press forward without congressional buy-in on issues such as immigration, while taking a more aggressive tack in blaming the GOP for Washington's gridlock.

On Tuesday, Mr. Obama appeared to taunt Republicans for a lawsuit that House Speaker John Boehner (R., Ohio) has threatened to file to challenge the president's use of executive authority. "Middle-class families can't wait for Republicans in Congress to do stuff," Mr. Obama said during a speech along Washington's waterfront. "So sue me. As long as they're doing nothing, I'm not going to apologize for trying to do something."

A spokesman for Mr. Boehner rejected the president's assessment. "House Republicans have passed over 40 common-sense jobs bills," said Michael Steel. He said "Democrats, led by President Obama," had blocked help to Americans.

This week, Mr. Obama conceded that his top second-term priority—overhauling immigration laws—is dead for now. And on Monday, the Supreme Court whittled away part of the president's signature health-care law, allowing closely held companies to invoke religious objections to opt out of covering contraception for their employees.