As many revenge dramas, Shakespeare`s A Winter`s Tale is set in a fictitional universe, oddly blending geographically distant Italy with pagan Graeco-Roman culture. For a king to consult the oracle of Delphi before taking a major action would have been normal procedure in 5th or 4th century Athens; the oracle itself and consultation with pagan oracles would not have been done in Renaissance Europe (though people might pray to saints or make pilgrimages to holy Christian sites).
In part, the consultation with the oracle shows the influence of Seneca, a Roman writer, on the genre of revenge tragedy; such consultations were common in both Seneca`s plays and the Euripidean tragedies that influenced them.
Christian tradition does include inescapable prophecies, by primarily Jewish prophets, e.g. the prophecy of the Magi concerning the birth of the King of the Jews leads to Herod the Great`s Massacre of the Innocents in the Gospel of Matthew. Prophecies by Jeremiah etc. are an important part of the Judaeo-Christian tradition, but the Delphic oracle is a literary device for Shakespeare (like the mention of Persephone in your other question) invoking pagan literary generic conventions.