Francesca Hogi, 40, had settled into her aisle seat for the flight from New York to London when the man assigned to the adjoining window seat arrived and refused to sit down. He said his religion prevented him from sitting beside a woman who was not his wife. Irritated but eager to get underway, she eventually agreed to move.

Laura Heywood, 42, had a similar experience while traveling from San Diego to London via New York. She was in a middle seat — her husband had the aisle — when the man with the window seat in the same row asked if the couple would switch positions. Ms. Heywood, offended by the notion that her sex made her an unacceptable seatmate, refused.

“I wasn’t rude, but I found the reason to be sexist, so I was direct,” she said.

A growing number of airline passengers, particularly on trips between the United States and Israel, are now sharing stories of conflicts between ultra-Orthodox Jewish men trying to follow their faith and women just hoping to sit down. Several flights from New York to Israel over the last year have been delayed or disrupted over the issue, and with social media spreading outrage and debate, the disputes have spawned a protest initiative, an online petition and a spoof safety video from a Jewish magazine suggesting a full-body safety vest (“Yes, it’s kosher!”) to protect ultra-Orthodox men from women seated next to them on airplanes.

Some passengers say they have found the seat-change requests simply surprising or confusing. But in many cases, the issue has exposed and amplified tensions between different strains of Judaism.