This is a pernicious falsehood. It is the duty of all Catholics to publicly live their faith and it is especially the duty of the laity to live their faith in the public sphere.
This is first and foremost made clear in Sacred Scripture. Our Lord makes this explicit to His followers: "You are the light of the world. A city seated on a mountain cannot be hid. Neither do men light a candle and put it under a bushel, but upon a candlestick, that it may shine to all that are in the house. So let your light shine before men, that they may see your good works, and glorify your Father who is in heaven" (Matthew 5: 14-6; Cf. Mark 4: 21; Luke 11: 33-6). Similarly, St. James tell us in his epistle that "faith without works is dead" (James 2: 20, 26).
One can find this clear teaching reiterated by the Church. I do not know what the so-called "spirit" of Vatican II says about it, but the texts of Vatican II explicitly state that it is the vocation of the laity to carry their faith into the world through their actions:
Each individual layman must stand before the world as a witness to the resurrection and life of the Lord Jesus and a symbol of the living God. All the laity as a community and each one according to his ability must nourish the world with spiritual fruits. They must diffuse in the world that spirit which animates the poor, the meek, the peace makers-whom the Lord in the Gospel proclaimed as blessed. In a word, "Christians must be to the world what the soul is to the body" (Lumen gentium, 38).Thus it seems clear that no Catholic can be faithful to the teachings of the Church while holding that his faith is merely a private matter. The Church explicitly teaches otherwise. Faith cannot be something we simply hold privately. Rather, it should be the source of our actions, giving birth to hope and charity, and motivating us to conform all the things and actions in our lives, whether they be public or private, to Christ.