The cut happened after an at-bat in Sunday's series finale against the Yankees. The bat rack has a metal top with some bolt heads that rise out of the top, and Martinez sustained his cut on the top of his right thumb against the metal part of the rack.
The Tigers were off on Monday.
Andy Dirks took Martinez's place in the lineup as the designated hitter, batting fifth behind Prince Fielder. Don Kelly entered the starting lineup as the left fielder, batting eighth.
Martinez had a slow opening week at the plate. He showed no signs of rust in Spring Training, after he missed last season recovering from knee surgery. Martinez's 3-for-21 opening week included an 0-for-4 performance Sunday against the Yanks. He flew out to shallow center in his final at-bat with runners at the corners and one out against reliever David Robertson.