While the Catholic Church, as far as I am aware, has no official teaching about the rebuilding of the Temple, there are solid theological arguments to suggest that in the plan of God there is no need for a new Temple.
The principal reason for the Temple was to have a dignified place in which to offer sacrifices to God. When Moses received the Ten Commandments from God, the Israelites were to offer two sacrifices of a lamb each day, one in the morning and one in the afternoon (cf. Ex 29:38-39). These were offered from that time on, wherever the Israelites were, until Solomon built the first Temple on Mount Moriah in present-day Jerusalem around the year 957 BC. From then on it was there that the sacrifices were offered on behalf of the Jews scattered throughout the world.
When Solomon’s Temple was destroyed by Nebuchadnezzar in 586 BC, it was the will of God that it be replaced, so that the sacrifices could once again be offered. Thus, when the Israelites were allowed to return to Jerusalem from exile in Babylon, they rebuilt it, completing the work in 515 BC.
In the plan of God, the sacrifices of lambs offered each day in the Temple prefigured and were replaced by the sacrifice of Christ, the Lamb of God, on the Cross. His one and only sacrifice is made present in the Mass, which is offered every day in churches throughout the world. So, in the new and definitive covenant brought about by Christ, the Temple is no longer needed.
When Jesus breathed his last on the Cross, he said Kalah, “It is finished” (Jn 19:30). By those words he was referring not only to the end of his earthly life but also to the reason for his becoming man: to redeem mankind by his death on the Cross. His death brought to an end the old covenant and inaugurated the new. Indeed, when he instituted the Eucharist in the Last Supper, he said, “This chalice is the new covenant in my blood” (1 Cor 11:25).
We see this too in the fact that when Christ died on the Cross, the curtain in the Temple was torn in two from top to bottom, exposing the Holy of Holies, the most sacred part of the Temple (cf. Lk 23:45). The Holy of Holies symbolised heaven, indicating that heaven was now opened. At the same time, when the high priest in the Temple finished sacrificing the last lamb for the Passover that day, he too would say Kalah, “It is finished.”
The old covenant was finished, replaced and fulfilled by the new. The daily sacrifices in the Temple were replaced by the sacrifice of Christ on the Cross, made present in the Mass.
Confirmation of God’s will in this might be seen in what I wrote about the scarlet thread (cf. Question Time 6, q. 763). As related in the Jewish writings the Talmud and the Zohar, a scarlet thread used to be attached to a door of the Temple when the high priest entered the Holy of Holies on the feast of Yom Kippur to offer sacrifice in atonement for the people’s sins. If the thread turned white, the people understood that God was pleased with the sacrifice and they would rejoice.
But in the 40 years prior to the destruction of the Temple in the year 70, the thread never turned white (cf. Talmud, Rosh Hashanah 31b). Christ’s death took place around the year 30, forty years before the destruction of the Temple. This might be interpreted as meaning that after Christ’s death God was not pleased with the Jewish sacrifice because it had been replaced by the sacrifice of Christ on Calvary.
Many Jews believe that when the Messiah comes, he will build the new Temple. Indeed, Christ, the Messiah, has come and he is himself the new Temple: “Destroy this Temple and in three days I will raise it up.” Christ, the new Temple will remain forever, among other ways in the Eucharist.