My God, fasting and abstinence is firmly rooted in the Old and New Testaments. Exodus, Daniels, Numbers and mention the practice of fasting and abstinence. Jesus fasted 40 days in the desert and taught his disciples to fast. So the fact that the Catholic Church has rules for fasting and absintence during certain times of the year as a sign of obedience and respect is certainly not stretch.
And by the way, fasting is not Catholic Doctrine....it can change over time...it is a practice....and where does it say in the Bible that everything we need to believe in is in the Bible?