I've had UA not check my passport on many occasions. First of all, you had to have flown UA with a passport in the recent past.
Secondly, you have to check-in online.
Third, you have to either carry your bags with you or check them at curbside. Sometimes, you can check bags inside, and simply be asked for an ID, but not your passport. Sometimes, they want to see the passport.
Once, I was traveling with a friend and we were going to Hong Kong, Singapore, Bangkok and back. The night before, I asked "You do have your passport," to which the reply was - "Yea, I used it just the other day - it's in my pants." Well, it was - and the pants got washed. I mean it was really ruined. Technically, it was all there, but there were blue splotches and stuff on all the pages. Not to mention that it was now three dimensional, not flat.
I thought about it a while and I said, "Well, let's try it." We checked in online, went to the airport, checked bags, showed them our driver's licensees, and off we went. No one ever asked to see our passports until we got to Hong Kong. Passport control looked at it - asked what happened to it - he said he washed it - they stamped it and that was it. This continued the same in the other two counties as well. Only coming back into the US did we have a problem, and even then the customs guy just said, "you know you need to replace this before you travel again." Duh.
So, traveling without a passport - No. Traveling without UA looking at it - sometimes.
Billy