Gene Wolfe’s novel PIRATE FREEDOM has a protagonist who is a modern Catholic Priest, but who was also a pirate on the Spanish Main 400 years ago, when he walked out of the novitiate his mafioso father enrolled him in our near future, and who then found himself stuck in the past…it’s a good read. Also read his wonderful short story THE DETECTIVE OF DREAMS, which moves from a hilarious pastiche of “consulting detective” mysteries of the early 20th century, through deep mystery, and finally to joyful transcendence.
Walter M. Miller’s A CANTICLE FOR LEIBOWITZ centers around a monastery in Arizona after a post-nuclear holocaust. Like the Irish monks who preserved the texts of the Greeks and Romans, the monks of the Order of St. Leibowitz faithfully preserve the scientific and technical texts of the 20th century through a new era of barbarism.
James Blish’s Hugo Award-winning A CASE OF CONSCIENCE follows Fr. Ruiz Echeverria, a Jesuit geologist and team member on a planetary survey team on the planet Lithia, who must deal with the scientific and theological puzzle of first contact with an intelligent, gentle alien species who manifest all the virtues, but have no conception of God.
Mary Doria Russell’s THE SWALLOW and its sequel CHILDREN OF GOD follows a team of Jesuits to an ultimately tragic confrontation with the two intelligent species of an alien planet, and explores the mystery of the Problem of Suffering.
R.A. Lafferty’s PAST MASTER’s protagonist is St. Thomas More himself, who is brought forward into the far future to help solve the problems of a planet whose political system was based on St. Thomas More’s own SF novel, UTOPIA.
William Peter Blatty’s THE EXORCIST focuses on the spiritual travails of Fr. Damien Karras, a Jesuit psychiatrist who is struggling to maintain his faith and is approached about the possibility of an exorcism of a young girl…
F. Paul Wilson’s vampire novel MIDNIGHT MASS depicts a world where vampires have gained control, and a New Jersey priest fights back with a small band of others.
Evelyn Waugh’s BRIDESHEAD REVISITED follows a young man of no particular religious inclination who finds himself increasingly drawn to an English Catholic family through a set of tragic circumstances, and finds how Divine Grace and Love can manifest itself through all the varieties of love.
Anthony Burgess’s cold war era spy novel TREMOR OF INTENT follows two former friends in an unnamed British intelligence agency, whose bond was forged by their Catholicism within an often hostile British public school system. See also Tim Powers’ cold war fantasy novel DECLARE, which combines a Catholic protagonist, real life cold war figures, and a demonic incursion into our world.
In BROTHER PETROC’S RETURN by “S.M.C.” (who was actually Sister Mary Catherine, a Dominican nun), a monk who died in 1549 returns by a miracle to 1949, where he confronts the age of modernity.
Fr. Robert Hugh Benson wrote a number of novels with Catholic protagonists, including LORD OF THE WORLD (recently mentioned by Pope Francis), THE NECROMANCERS, BY RACK AND BY ROPE, and the collection of supernatural stories A MIRROR OF SHALOTT, where a group of priests relate the most unusual experience they have ever had around the fireplace after dinner.
Michael Flynn’s EIFELHEIM depicts a Catholic priest who encounters (and converts!) an alien expedition to earth in the middle ages. Poul Anderson’s THE HIGH CRUSADE has its medieval English Catholic characters, including a sympathetically depicted priest, gain control of an alien starship after they unexpectedly defeat its crew, which was bent on subjugating earth.
Dean Koontz’s post-conversion novels, especially his ODD JOHN series, often have Catholic protagonists.
Brian Moore’s novel CATHOLICS (later filmed with Trevor Howard and Martin Sheen) is set in the near future when the Latin Mass has been proscribed, and a liberal American priest is sent to a remote Irish monastery which has defied the proscription.
, there’s Andrew Greeley and his Father Blackie books. These are highly not recommended. 
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My characters are Catholic, too!