The House now plans to adjourn Wednesday for the monthlong summer recess.
The bill urges the VA to define military sexual assault as a service-connected cause of PTSD.
The legislation comes in response to a string of high-profile security incidents.
A gunman shot 11 people in a Lafayette, La. movie theater last week.
Shooting suspects in South Carolina and Louisiana bought guns despite questionable legal histories.
The bill would prevent domestic violence perpetrators from obtaining guns.
House Majority Leader Kevin McCarthy outlined next week's schedule.
The bill, passed 241-179, would withhold certain federal law enforcement grants.
The bill would keep states from issuing mandatory labeling laws for foods that contain GMOs.
The Speaker said "it’s pretty clear" that a stopgap bill will be needed in September.