Synge's Riders to the Sea is definitely a tragedy.
Fate is the governing force in Riders, as it often is in Greek tragedy. Maurya and her family have only one means of survival--the sea. The men are doomed because the same sea they must travel to earn a living will sooner or later destroy them. Maurya is trapped by forces she cannot control and cannot even understand.
Maurya suffers through the loss of nine males, including her husband and all of her sons, and she only reaches any kind of resolution or peace when she finally has nothing left to lose.
The family is, in effect, wiped out by a force beyond its control. Notice the weaving occurring when the play opens--this is in imitation of and an allusion to the Greek fates who weave human destiny. Notice how Michael's fate is determined by how a sock was stitched.
Synge mixes Irish myth with Greek myth to form his tragedy, but for your purposes, the above should establish that the play is a tragedy in the Greek tradition.