How do Catholics respond to the claims that the cross has pagan origins? According to An Expository Dictionary of New Testament Words (London, 1962), W. E. Vine, p. 256.  :


  The shape of the [two-beamed cross] had its origin in ancient Chaldea, and was used as the symbol of the god Tammuz (being in the shape of the mystic Tau, the initial of his name) in that country and in adjacent lands, including Egypt. By the middle of the 3rd cent. A.D. the churches had either departed from, or had travestied, certain doctrines of the Christian faith. In order to increase the prestige of the apostate ecclesiastical system pagans were received into the churches apart from regeneration by faith, and were permitted largely to retain their pagan signs and symbols. Hence the Tau or T, in its most frequent form, with the cross-piece lowered, was adopted to stand for the cross of Christ.”


There are many other references that agree that prior to 3rd or 4th century only non Christian groups ever used the symbol of the cross in their worship. There is also the matter of whether or not Jesus was put to death on a cross or a single upright stake as the Ancient Greek words stauros indicates. 

Thus I am asking: How does the Catholic Church explain the cross becoming acceptable to use and venerate as a symbol of Christianity?
This is a non-starter because it rests upon multiple logical fallacies, meaning it's a non-argument for anything other than one's inability or unwillingness to be logical.

Namely, it commits the Genetic Fallacy: 'x has origins in somone or something evil, therefore, x must be bad in and of itself' (e.g. 'circumcision is evil because pagans invented circumcision').

It also commits the Association Fallacy (Guilt By Association): 'x was done or caused by evil [person] y, and therefore all instances of x implies the evil [of person] y' (e.g. 'pagans used crosses on which to torture people, therefore anything plus sign or cross shaped is evil' or 'all guns are evil because some people shoot humans with them.')

There are probably a few more, but one is sufficient, and you get the point.

But more simply, the crucifying Him on a cross was an evil done to Christ. Just as killing Him was. But the evil done to Him was to our benefit (Gal 3:13; 2 Cor 5:21 etc.).

1 Cor 1:23-24: the cross is a stumbling block and an apparent defeat only to those who are perishing (2 Cor 4:3), but to us it is our salvation.

Gen 50:20: the Romans and Jews putting Christ to death was an evil objectively, but subjectively, we recieve benefit because of it.

Also, God can sanctify that which is pagan (see the above logical fallacies upon which some rest to object to this fact) for His own purpose. An example is circumcision. God doesn't care what other's signified or meant by, since why would He? Only He matters. Romans used the cross to torture criminals. So what? God used the cross to present His salvation to the world, and by definition becomes something holy, in spite of itself.