The Catholic Church has never really "given up", but there are times when it has lost interest because there was effectively nothing it could do to break the status quo.

People sometimes forget that the two Churches actually agreed to a reconciliation at the Council of Florence in 1445, an agreement which was even approved by the Emperor. The response in Constantinople was, to put it mildly, unfavorable. The possibly apocryphal chant was "Better the Turkish turban than the Papal tiara!" They got their wish 8 years later.

Looking back, the greatest problems with reconciliation may not have actually been doctrinal stubbornness, though that certainly helped. The great schism was a formalization of an existing situation, but it was more one of papal-imperial relations manifesting through and alongside doctrinal disagreements. However, it was really the mutual animosity that built up in the West from the first two crusades and in the East from the Fourth Crusade and the Latin Byzantine state which seems to have taken over 700 years to overcome.