WASHINGTON—President Donald Trump on Monday gave his strongest endorsement yet to Alabama Republican Senate candidate Roy Moore, linking his candidacy to the need to pass tax cuts, criminal-justice measures, stricter immigration laws and enforcement, and more.

Mr. Trump’s statement in an early-morning tweet marked the first time the president has backed Mr. Moore by name since the former judge’s candidacy was rocked by allegations of sexual misconduct involving teenage girls decades ago, which he has denied.

“Democrats...