The 34-year-old former England international picked up an Achilles injury on the pre-season tour of Norway soon after joining the club.
Last weekend he had surgery on the problem for the third time and Blackwell admits Stone may never get to make his Whites debut.
He told the Yorkshire Evening Post: ‘There’s a possible career-threatening situation here. There’s no timescale and, in my mind, I’ve had to write him off for the rest of the season. Anything I get back from that is an absolute bonus.’