The airline declared an "operational emergency" at several maintenance bases last month, telling scheduled mechanics to show up at work or risk termination.
The Aircraft Mechanics Fraternal Association, which represents some 2,400 Southwest mechanics, said in a statement that its technicians are obligated by law to "protect everyone who steps on that aircraft and there is zero evidence presented of a non-legitimate discrepancy" by the mechanics.
More Southwest flights have been canceled since mid-February compared with competitors. The airline has said it usually plans to have about 20 planes out of service for unexpected maintenance issues but that number has doubled.
On Tuesday, 89 Southwest flights were canceled, about 2 percent of its schedule, according to flight-tracking site FlightAware. In comparison, JetBlue had 12 canceled flights and United had eight.
"Customers are harmed. Our employees suffer through reduced profit sharing," Kelly said at a transportation conference hosted by J.P. Morgan Chase. A spokesman for the airline declined to provide a more specific estimate of how much Southwest is losing from the issue. Analysts expect Southwest to post revenue of about $5.3 billion this quarter.
Kelly's comments come a day after Southwest started selling flights to Hawaii from California, after the airline received federal approval last month to operate longer, over-water routes with its Boeing 737s.
Kelly later said in an interview with CNBC's Closing Bell on Tuesday that the planes are also able to fly from North America to the United Kingdom but said that destinations there are not currently among the airline's plans. The airline is interested in developing so-called code shares with other airlines, which would allow Southwest to fly one segment of a route and be able to sell tickets on another leg that is operated by another carrier, in order to widen its route network.