WASHINGTON — Several lawmakers from both parties on Wednesday accused President Obama of violating the War Powers Resolution by continuing American participation in NATO’s air war in Libya without Congressional authorization, but they struggled with the question of what Congress can or should do about it.

At a hearing of the House Foreign Affairs Committee, several members attacked Mr. Obama for failing to withdraw United States military forces from conflict after the expiration of a 60-day deadline for hostilities that have not been approved by Congress. The Libyan operation reached that deadline, which was imposed by the war powers law of the Vietnam era, on Friday.

“The president is not a king, and he shouldn’t act like a king,” said Representative Dan Burton, Republican of Indiana. Representative Brad Sherman, Democrat of California, said the administration was treating lawmakers as “irrelevant” by failing to acknowledge that the deadline had passed or to explain itself....