What legislation required a majority of a state's prewar voters to swear loyalty to the union before restoration could begin?

 The Wade-Davis Bill  (1864)

Detail:

President Abraham Lincoln had proposed lenient approaches to Reconstruction.  Lincoln's plan would have required only 10% of a Southern state's voters to pledge loyalty to the Union, as well as recognizing the freedom of slaves.  

Senator Benjamin F. Wade (from Ohio) and Representative Henry W. Davis (from Maryland) proposed their bill in February, 1864, requiring at least 50% of a state's prewar voters (white males) to swear loyalty to the Union, as well as giving blacks the right to vote.  The measure passed Congress, but Pres. Lincoln did not sign it.  (That's called a "pocket veto.")  So for the time being that measure died.

After Lincoln was assassinated in 1865, the "Radical Republicans" in Congress (who wanted harsher terms imposed on the South) tended to get their way.  President Andrew Johnson vetoed the stern Reconstruction Acts passed by Congress, but Congress overrode his vetoes and implemented their plans.  Tension between Congress and Pres. Johnson also led to the House of Representatives bringing impeachment charges against him.  The Senate narrowly failed to convict Johnson, so he stayed in office.  But Congress was clearly running the show.