This is a common sentiment, voiced by Protestants and atheists for centuries.
It can't be done.
Contrary to reports, the Vatican is not rich. It's annual budget is less than $300 million (Harvard's annual budget is $3.7 billion). It has an endowment of about $1 billion (Harvard has a whopping $30.7 billion endowment).
Treaty obligations prevent the Vatican from selling any artwork. Article 18 of the 1929 Lateran Concordat with Italy which created the modern Vatican State says:
“The artistic and scientific treasures existing within the Vatican City and the Lateran Palace shall remain open to scholars and visitors, although the Holy See shall be free to regulate the admission of the public thereto”If it sold its property or artwork (each valued at 1 Euro), it would lose its sovereignty. Italy would be within its rights to take it all back.
But the above two reasons are silly secular reasons. The real reason the Catholic Church cannot sell the artwork is simple: God is Truth, Goodness and Beauty.
The artwork is a necessary tool to evangelize people to understand God. Asking the Catholic Church to sell the artwork is very much like asking the Catholic Church to burn every copy of Scripture in the world and never mention Jesus Christ again. The request substantially asks the Church to hand over its witness to God to the highest bidder. It requests that the Church henceforth refuse to effectively witness to God.
It cannot be done.
Any person who even entertains the idea of selling of Vatican art, by the very fact that he proposes it, demonstrates that he knows absolutely nothing about the Gospel or the Catholic Faith.
According to even secular sources like The Economist, the Catholic Church is the single largest charitable organization in the world. We are charitable because we know Christ. If we lost the art, it would be harder to know Him, harder to remember our purpose. In charity, the Church must keep her image of God always before Her, and that image is BEAUTY.