Catalyzes the reaction that attaches fucose through an O-glycosidic linkage to a conserved serine or threonine residue in the consensus sequence C1-X-X-S/T-C2 of thrombospondin type I repeats (TSRs) where C1 and C2 are the first and second cysteines of the repeat, respectively. O-fucosylates members of several protein families including the ADAMTS, the thrombospondin (TSP) and spondin families. GDP-beta-L-fucose + L-seryl-[protein] = 3-O-(alpha-L-fucosyl)-L-seryl-[protein] + GDP + H(+) GDP-beta-L-fucose + L-threonyl-[protein] = 3-O-(alpha-L-fucosyl)-L-threonyl-[protein] + GDP + H(+) Does not require divalent metal ions for optimal activity. Protein modification; protein glycosylation. Mainly located in the endoplasmic reticulum. Belongs to the glycosyltransferase 68 family.