WASHINGTON—Senate Republicans’ plan to extract more savings from Medicaid as part of their revised tax and spending package triggered a backlash within the party, raising concerns that leaders could miss their self-imposed deadline of getting the bill to President Trump’s desk by July 4.

The clean energy tax credits stem from former President Joe Biden’s climate law, considered a landmark for the United States’ clean energy transition.

A federal judge on Monday ordered the government to restore funding for hundreds of National Institutes of Health grants cut by the Trump administration, ruling that the government discriminated again...