Thalassia testudinum does possess isolated crystalline silica deposits. Some seagrass fossil, similar to the extant T. testudinum , from Florida also showed the same trait, except they were calcium oxalate crystals. There is a very good publication on it entitled Biomineralization in Thalassia testudinum (Liliopsida: Hydrocharitaceae) and an Eocene Seagrass which you might look into.
Phytoliths are microscopic silica (Si(OH)4) structures occurring in plant tissues (Piperno 2006: 9-15). These inorganic deposits form the different cellular structures of the plant and remain after the organic materials decay.However, I am not very sure about seagrass