“This is going to quickly emerge as one way to relieve the demand pressures that are being put on the system,” said Ceci Connolly, managing director of the PriceWaterhouseCoopers Health Research Institute. “We’re just getting started.”

Texas is among a handful of states that still require an in-person exam before a telemedicine consult can take place, according to the American Telemedicine Association, a trade group in Washington. Other states have vaguely worded policies that are not clear on whether an in-person visit is needed first.

In a recent interview, Mr. Gorevic said Teladoc’s national network of 700 board-certified physicians receive special training in how to conduct telemedicine consultations and follow clinical guidelines that are “specific to telehealth.” The company, which operates in every state except Arkansas and Idaho, has seen rapid growth in demand for its services, he said, with almost 300,000 consultations last year, double the number in 2013. It has more customers in Texas than in any other state.

The company says its doctors are available 24 hours a day, seven days a week. When a customer requests a consultation by calling or going to the company’s website, Mr. Gorevic said, a doctor licensed in the person’s state responds within 10 minutes on average. Its doctors do not prescribe narcotics, other controlled substances, or what it refers to as “lifestyle drugs,” such as Viagra.

Teladoc’s conflict with the medical board started in 2011, when the board threatened to discipline the company’s doctors for prescribing drugs to patients they had not seen in person. Teladoc sued, accusing the board of changing its rules without going through the proper process. Late last year, a state appeals court sided with Teladoc, which led the medical board to issue an emergency rule in January. Teladoc won a temporary injunction while the board went through its formal rule-making process over the last few months.

The new restrictions are to take effect in June. Before voting on them, Frank Denton, an investment executive from the Houston area who was the only board member to oppose them, said telemedicine was the “least desirable” way to deliver health care but that “the public should have the right to use that option.”

But another board member, Dr. George Willeford of Austin, expressed the opinion of the majority when he said he worried that telemedicine, unless properly regulated, could weaken doctor accountability to patients. “I’m terribly, terribly worried about the absence of responsibility and accountability,” he said.