The 20-year-old had been on trial at Brunton Park after being released by the Scottish champions at the end of last season.
"We're not going to take Robbie Thomson," Abbott told Carlisle's official website. "He's got other options at the moment and he will go off to pursue those.
"We looked at him as a possible permanent signing, but we weren't sure about him challenging [first-choice keeper] Mark Gillespie.
"We'll keep looking for another goalkeeper, but it will probably be a loan deal if we can make that happen."