Watson, 26, suffered the injury in the North Riding Senior Cup win over Guisborough Town in midweek.
He was substituted towards the end of the first half after taking a knock. It initially wasn't thought to be too serious, however the news of the break has been confirmed.
It means Watson will be unavailable to face his former club Farsley Celtic in the EVO-STIK League Premier Division on Saturday.
Keep up-to-date with our exclusive email newsletters.
Subscribe