While cutting fares, United is also adding capacity, said Virgin America CEO David Cush. "[United] has flown these routes with eight flights a day for many, many years. They're going up to 14 flights a day in June. Again, very positive for the consumer. Good for fares. But a little bit dangerous bottom line-wise."
"It took us four years to get into Newark. The slots were locked into the legacy carriers," Branson said. "It was only when American [Airlines] went bankrupt that we actually managed to get some slots there."
For its part, United told CNBC, "United connects Newark Airport and the New York area to more destinations around the world with more flights than any other airline. We offer the most frequent service from Newark to Los Angeles and San Francisco and a range of options for business and leisure travelers, and we welcome the new competition."
The new competition from Virgin comes a day after United—the largest carrier in New Jersey and New York—announced a 20-year extension of its Newark Airport lease and plans to spend $150 million on improvements there.
United Continental, Delta, and what's going to be the combined US Airways-American Airlines are the big three carriers in the United States, following the industry's recent round of consolidation.
(Read More: Airbus Comes to US, Puts More Pressure on Boeing)
"Let the other guys consolidate. It's a much more efficient industry structure," Cush said. "[But] we need to make sure there's an environment where new entrants can still come in."
Cush added, "The biggest issue is we don't have new airlines starting up. We're the only airline still in business since JetBlue in the domestic U.S. And there are no new airlines on the board."