WASHINGTON—Senate Republican leaders left members from both ends of the party unsatisfied with their version of President Trump’s “big, beautiful bill,” touching off a fast round of new negotiations to hit their self-imposed deadline of putting the bill on his desk by July 4.

Democratic Sen. Tina Smith of Minnesota confronted GOP Sen. Mike Lee of Utah at the Capitol Monday over a social media post on the Minnesota lawmaker shooting.

Smith added that Lee “seemed a little surprised to be confronted” and told political commentator Brian Tyler Cohen that she effectively had to chase him down because he had his phone to his ear and wa...