Based on the dialogue found in act 5, scene 2 of Shakespeare's Romeo and Juliet, we can see that, unfortunately, Friar John did not try to deliver the letter to Romeo once Friar John was released from quarantine.
However, Friar John did try desperately to send the letter while he was in quarantine. He tried to get messengers to either carry the letter to Romeo or to return the letter to Friar Laurence, but the authorities were so afraid of spreading the pestilence, which was probably the Black Plague, that they forbade the letter to leave the house of quarantine, as we see in his lines:
I could not send it,--here it is again,--
Nor get a messenger to bring it thee,
So fearful were they of infection. (V.ii.14-16)
In this passage, Friar John is returning the letter to Friar Laurence, probably because he knows it's too late to deliver it, and returning it is the best he can do. As a result of Friar John being thwarted from delivering the letter, Romeo is of course led to believe that Juliet truly died, which, sadly, led to his choice to commit suicide followed by her own suicidal decision as well.