Since the Catholic Church has been in existence for 2000 years, the answer depends on the when, where, how and why of the teaching, since different levels and emphasis have been disseminated over the 2000 years. The best way to understand it is to divide the Catholic teaching into "official teaching" and "unofficial teaching."Â "Officially," the Catholic Church does continue to teach geocentrism since all patristic teaching and magisterial teaching in the Catholic Church is timeless, that is, there is no time limit that the teaching is only good for a certain period.
The Church Fathers were in consensus that geocentrism was a biblical and ecclesiastical doctrine, and the Council of Trent said that whenever the Fathers are in consensus we are obliged to believe their teaching; the medievals said the same (led by Thomas Aquinas); and the Church herself said in 1616 under Paul V that Copernicanism was "formally heretical," which judgment was reiterated and enforced at the trial of Galileo in 1633 under Pope Urban VIII. These decisions were disseminated throughout Europe, in all its universities and to all its papal nuncios. This decision was reinforced by Pope Alexander VII in 1664 in his encyclical, Speculatores Domus Israel, affirming all the books that were put on the Index of Forbidden Books, including Copernicus' De Revolutionibus; Kepler's Epitome Astronomia Copernicae, and Galileo's Dialogo, were condemned because they taught heliocentrism. Although these books were taken off the Index in 1835 under Pope Gregory XVI, there was no rescinding of the doctrine of geocentrism from 1616 and 1633, and no endorsement of Copernicanism. The significance of this is that in 1775, French astronomer, Jospeh Lalande, went to the Vatican to seek to remove Galileo's name from the Index, but he was told by the Holy Office that it could not be done unless the verdict of Galileo's trial was rescinded. It has not been rescinded until this very day.
Two other significant things occurred:
First, in 1822, Canon Settele was permitted by the Vatican under Pius VII to publish a book defending heliocentrism. Settele had been denied by the previous pope. But during the reign of Pius VII, the whole process of getting the imprimatur for Settele was done by subterfuge. To start things off, the prefect of the Apostolic Office, Fr. Filippo Anfossi, refused to give Settele an imprimatur, citing the 1616 and 1633 judgments by the Church against heliocentrism, but Cardinal Maurizio Olivieri, who believed in heliocentrism, appealed to Pope Pius VII to override Fr. Anfossi's judgment. In order to convince Pius VII, Cardinal Olivieri lied to Pius VII, claiming that the 1616 and 1633 magisterial decisions did not condemn heliocentrism but only the kind of heliocentrism that Galileo was promoting, that is, one without elliptical orbits for the planets. This was a bald-faced lie because the 1616 and 1633 magisteriums never even discussed the issue of elliptical orbits, much less based their decision on them. The magisterium was crystal clear that the Earth didn't move. But Pius VII, after having been incarcerated in Florence by Napoleon from 1809 to 1814, became a weak and sickly pope, open to being manipulated. Moreover, since Napoleon had seized all the Galileo records from the Vatican in 1809 and put them in a library in Paris, neither Fr. Anfossi nor Pius VII had any recourse as to what transpired in the 1616 and 1633 decisions of the Vatican against Galileo, which made issuing the imprimatur to Settele much easier for Cardinal Olivieri. Since the records were gone, Olivieri could make up any story he wanted and convince Pius VII it was true, which he did.Â
Second, when Isaac Newton published his famous book in 1687, The Principia Mathematica, it taught heliocentrism as proven by science. But the Catholic editors of Newton's book, Jacquier and LeSeur, put a disclaimer on the book stating, "Newton in his third book assumes the hypothesis of the earth’s movement. The author’s propositions could not be explained except on the same hypothesis. Hence we have been obliged to put on a character not our own. But we profess obedience to the decrees made by the Supreme Pontiffs against the movement of the earth." This showed that despite Newton's views, the Catholic Church was adhering to its traditional teaching of a fixed Earth.There has been no change in the Catholic Church "official" teaching for geocentrism and against heliocentrism. The last address on the subject was made by John Paul II in 1992 in which he appeared to side with Galileo, but since this was not an official teaching of the Church but only an private address to the Pontifical Academy of Science, it can only be considered the pious opinion of the pope but not an official teaching of the Catholic Church.Â
Hence "unofficially," the Catholic Church today does not appear to be teaching geocentrism, but even that conclusion is in question considering that in 1990 Cardinal Joseph Ratzinger, who at that time was the Prefect for the Congregation of the Doctrine of the Faith, gave a speech in Parma, Italy, stating that geocentrism was a viable scientific position since Newton's mechanics had been replaced by Einstein's relativity, and that the Church's decision against Galileo was right and just.
This is a question to ask of people with the other point of view, not geocentrists. But here it is: the course of the Earth is a helix, like a spring or spiral staircase, as the entire solar system moves and the Earth moves around the sun. Since there is no friction in space and the entire system is moving at a constant speed and bound together by gravity, there's no problem with the Earth "following" or "being pulled along by" the Sun.