Thompson, who shot a first-round 73, ended on an even-par 142 with the cut set at three-under.
Cameron Champ and Lanto Griffin shared the lead on 12-under par, a stroke ahead of Sweden's Henrik Norlander.
Thompson, a 28-year-old Floridian, missed a six-foot putt for birdie on her final hole, which would have matched the lowest score on the PGA Tour by a woman.
Michelle Wie West shot 68 in the 2004 and 2006 Sony Open in Hawaii.
Thompson was looking to become the first woman to make the cut on the PGA Tour since Shirley Spork achieved a 105th-place result at the 1952 Barracuda Championship.
"I'm very proud. I played really well today, came out super early, and bogeyed 17 but made a great save on 18 there, and just overall played very steady," she said.
"As the day went on, I tried to stay committed to my targets out there and to my swing thoughts, and just enjoyed the whole experience," added Thompson, who was playing on a sponsor's exemption.
After play finished early due to bad light on Thursday, Thompson had to come back out to finish off her last two first-round holes - making a bogey on the 17th and a par on the last.
After a bogey at the 10th hole to begin her second round, she made three birdies before the turn and then put the cut within her sights after successive birdies on the first and second.
But bogeys on the par-3 fifth and eighth holes undid her hopes.
She went out of bounds with her drive on the fifth, being forced to take a penalty, and then two-putted on the eighth.
"I knew once I had made a few birdies in a row that I would be somewhat close, but I knew the cut line would keep on going lower and lower with these guys, especially with not that much wind out there," she said.
"I just really tried to stay focused on my game and focused on my swing thoughts, picking small targets and staying 100 percent committed, and whatever happens, happens."
Thompson, who won her sole major at the 2014 Kraft Nabisco Championship, last took an LPGA title at the 2019 ShopRite Classic.
Her most recent title came in a Ladies European Tour event in New York in October 2022.
She enjoyed plenty of support from the crowd at TPC Summerlin in Las Vegas but said the attention from fans and the media hadn't added to the pressure.
"It was more energizing. Seeing the people out there and hearing the cheers and seeing all the little kids, that's what I play for," she said.
"No matter what I'm shooting, I could shoot 80, and they'd be like, you did great. That's what it's all about, just inspiring."
Champ made seven birdies and three bogeys in his round of 67 after sparking in the opening round with a 63.
Griffin was also unable to match his 64 from Thursday with a 66 in his second round, but Norlander sparkled with his 64, ending with an eagle on the par-5 9th hole where he drilled his second shot 267 yards, before nailing a ten-foot putt.
