For example, in Act I, Scene i, Horatio refers to the ghost of Hamlet's father as a "spirit" and a "ghost." This suggests that he believes in the afterlife, which is a Christian belief. Additionally, in Act III, Scene iv, Horatio says, "I am more an antique Roman than a Dane," which could be interpreted as a reference to the Christian faith, which was widespread in the Roman Empire.
However, it is also possible that Horatio was not a Christian. For example, in Act I, Scene v, Horatio says, "There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy," which suggests that he is open to the possibility of other religions or spiritual beliefs. Additionally, in Act V, Scene i, Horatio says, "If there be any good thing to be done, / That may to thee do ease, and grace to me, / Speak to me," which could be interpreted as a reference to a non-Christian prayer.
Ultimately, it is impossible to say for certain what Horatio's religion was. However, there is evidence to suggest that he may have been a Christian, as well as evidence to suggest that he may not have been.