The History of Middle Earth (Vol 12) indicates that Aragorn and Arwen lived long and prospered in their roles as King and Queen of Gondor and that their children were accepted as the heirs to the throne.
King Elessar [Aragorn] and Queen Arwen reigned long and in great blessedness;
but at the last the weariness came upon the King, and then, while
still in vigour of mind and body, he laid himself down after the
manner of the ancient kings of Numenor, and died, in the hundred and
second year of his reign and the hundred and ninetieth year of his
After Aragorn's death, Arwen left Gondor and went back to her ancestral home. Again, there's a mention that her reign was a welcome one to the peoples of Gondor
But Arwen became a mortal woman, and yet even so it was not her lot to
die until she had lost all that she gained. For though she lived with
Aragorn for five score years after and great was their glory together,
yet at the last he said farewell and laid him down and died ere old
age unmanned him.
For the record, there are no indications of any rebellion or strife that occurred during their reign. If anything, his latter years as King were largely uneventful aside from some (highly successful) military campaigns to recover land that had been annexed by the Easterlings.