According to the Catechism of the Catholic Church,
1807 Justice is the moral virtue that consists in the constant and firm will to give their due to God and neighbor... Justice toward men disposes one to respect the rights of each and to establish in human relationships the harmony that promotes equity with regard to persons and to the common good. The just man, often mentioned in the Sacred Scriptures, is distinguished by habitual right thinking and the uprightness of his conduct toward his neighbor. "You shall not be partial to the poor or defer to the great, but in righteousness shall you judge your neighbor." "Masters, treat your slaves justly and fairly, knowing that you also have a Master in heaven."
According to Catholic Social Thought, one of the key principles of social moral theology is the so-called "universal destination of goods", for which the Catechism of the Catholic Church states:
2403 The right to private property, acquired or received in a just way, does not do away with the original gift of the earth to the whole of mankind. The universal destination of goods remains primordial, even if the promotion of the common good requires respect for the right to private property and its exercise...
2405 Goods of production – material or immaterial – such as land, factories, practical or artistic skills, oblige their possessors to employ them in ways that will benefit the greatest number. Those who hold goods for use and consumption should use them with moderation, reserving the better part for guests, for the sick and the poor.
Finally, according to St. Thomas Aquinas,
since injustice always consists in an injury inflicted on another person, it is evident that to do an injustice is a mortal sin according to its genus.
All these seems to mean that the rich (and more broadly, all those who "have more than they need") are obliged to share it with the poor, otherwise at the risk of committing a mortal sin. Is this a sound doctrinal conclusion from the matter at hand, according to Catholic doctrine?
Partially related question for reformed theology here.