AUSTIN, Texas—In a federal trial set to begin here on Monday, abortion-rights supporters are seeking to strike down a new provision of a state law that will require abortion clinics to qualify as "ambulatory surgical centers" starting next month, saying it will force even more of the state's facilities to close.

Since Texas Gov. Rick Perry signed a hotly contested law in July 2013 requiring abortion doctors to have admitting privileges at nearby hospitals, the number of licensed Texas clinics providing abortions...