The Giants are known to run a relatively tough camp that includes running laps for mental mistakes. Coach Joe Judge also had his team run sprints and do pushups as punishment for a teamwide melee earlier in the week to stress accountability.
But Judge noted that Looney's and Davis' decisions were more personal.
"These are tough decisions on guys. As you get vets later in their career, they started training camp, it's not uncommon for this to happen," Judge said Thursday. "Now, you don't want it to happen, but you have to respect the decision each person makes for their family. I'd say in both these cases, not to speak for the players, but both of them had family situations that they had to attend to, and that was a large part of the decision.
"I think both guys could've helped us. We left the door open for both guys based on what may open up in the future."
Fulton, 29, signed a one-year deal with the Giants this offseason that would have paid him $1.2 million. He received a $137,500 signing bonus.
A starter for most of his career, Fulton was battling for a roster spot with the Giants. He spent most of the early part of training camp working with the second-team offense. And even when starting left guard Shane Lemieux injured his knee, Kenny Wiggins took his spot ahead of Fulton.
Fulton was even seen this week at practice cross-training at center after Looney retired.
The news of Fulton's decision was first reported by Newsday.