WASHINGTON—President Trump backed further away from calls for stricter background checks for gun purchasers and emphasized mental illness instead as a prominent factor in mass shootings.

“I’ve said 100 times, it’s not the gun that pulls the trigger, it’s the person that pulls the trigger. These people are sick,” Mr. Trump told reporters Tuesday, echoing sentiments he has expressed in the past week, including during a political rally in New Hampshire.

After...