The worsening feud between Republic Airways Holdings Inc. and its pilots union highlights strains across the U.S. regional-airline industry, which is contending with a sudden pilot shortage just as business is being crimped by major carriers reducing commuter flights to some smaller airports.

Regional carriers, while far less well known than big mainline airlines, are enormously important to U.S. fliers. They carried 157 million passengers in 2013, the latest statistic available from their trade association, and were the sole source of flights at 431 U.S. airports.

Indianapolis-based Republic, the nation’s second-largest regional carrier by passengers, employs 242 planes to operate more than 1,200 flights a day on behalf of American Airlines Group Inc., United Continental Holdings Inc. and Delta Air Lines Inc. Republic has warned its 2,100 pilots that it could file for bankruptcy-court reorganization unless they accept a new contract that would significantly raise pilot pay—but not by as much as the union wants.

“Getting to a consensual agreement with the pilots is the best outcome,” Republic Chief Executive Bryan Bedford said in an interview. “But if we just can’t get there, at some point we have to surrender to that reality and go down a different path.”

The union, while sympathetic to the broader problems facing Republic, also argues that the airline reneged on some provisional agreements and flouted protocol by bringing its latest offer directly to pilots. Thus, the union declined to put it out to a membership vote.