The 28-year-old was hurt after being dismissed playing for Surrey in a T20 game against Hampshire on Wednesday.
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He will miss Friday's game against Glamorgan in Cardiff.
"I'm extremely embarrassed and apologetic to my team mates and fans for this moment of stupidity," Roy said. "Following my dismissal on Wednesday evening, I threw my bat into the ground purely out of self-frustration. It bounced, hit me, and as a result I sustained the injury.
He further said: "I've never been so motivated and so positive about my game, and I assure you I will learn from my mistake as I am my own biggest critic."
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