Was the theory of many extraterrestrial worlds considered blasphemous by the Roman Catholic Church during the 16th century?

Page 127 of For the Glory of God by Rodney Stark, not a Catholic, clarifies this.

From my investigation, the Catholic Encyclopedia (1907?) article Giordano Bruno is much closer to the mark:

Thus, his system of thought is an incoherent materialistic pantheism. God and the world are one; matter and spirit, body and soul, are two phases of the same substance; the universe is infinite; beyond the visible world there is an infinity of other worlds, each of which is inhabited; this terrestrial globe has a soul; in fact, each and every part of it, mineral as well as plant and animal, is animated; all matter is made up of the same elements (no distinction between terrestrial and celestial matter); all souls are akin (transmigration is, therefore, not impossible). This unitary point of view is Bruno's justification of "natural magic." No doubt, the attempt to establish a scientific continuity among all the phenomena of nature is an important manifestation of the modern spirit, and interesting, especially on account of its appearance at the moment when the medieval point of view was being abandoned. And one can readily understand how Bruno's effort to establish a unitary concept of nature commanded the admiration of such men as Spinoza, Jacobi, and Hegel. On the other hand, the exaggerations, the limitations, and the positive errors of his scientific system; his intolerance of even those who were working for the reforms to which he was devoted; the false analogies, fantastic allegories, and sophistical reasonings into which his emotional fervour often betrayed him have justified, in the eyes of many, Bayle's characterization of him as "the knight-errant of philosophy." His attitude of mind towards religious truth was that of a rationalist. Personally, he failed to feel any of the vital significance of Christianity as a religious system. It was not a Roman Inquisitor, but a Protestant divine, who said of him that he was "a man of great capacity, with infinite knowledge, but not a trace of religion."

As mentioned in one of the comments, Bruno was not condemned and executed for this scientific views. This proposition is a product of anti-catholic propaganda of 18th and 19th century (by Andrew Dickson White or William Draper - or the likes).

It is true that he took logic to Copernicanism and postulated existence of "innumerably many worlds" and again, by extension of existence of extraterrestial life. In general nothing in Catholic doctrine refuses that view - in fact, the opposite can be argued (as did Fr Angelo Secchi in 1879). Though it must be noted that there is no official position of CC on that issue - there are only personal beliefs of numerous hierarchs of the Catholic Church that it would be a waste of space if we were the only sentience in the universe.

However.

When one digs deeper into the facts on "Church war on science" trope, nothing seems as it's generally believed...

Copernicus was a Church cleric (canon) and his treaty was dedicated to the Pope. He was not persecuted and though his "De revolutionibus..." was officially forbidden by the Officium (i.e. Inqusition), it was due to - mostly - political and theological errors in his book, and once corrected (and owners of copies of his book were required to censor his book themselves), it was accepted by CC. [There is a question in SE about what exactly was censored and it is a fantastic read; go look it up if interested]

Same thing with Galileo. He was not persecuted, he was requested to either prove Copernican model or STFU. In addition, this was also centered around religious and doctrinal errors of Galileo and most notably, because he was a jerk who rubbed a lot of powerful people the wrong way. Anyway, as I like to point out, Galileo would do much better on his trial if he called as a witness one Johannes Kepler instead of Bible and his personal beliefs (i.e. he defended his theory using doctrine, not science). Unfortunately, Kepler was at that time extremely busy. Namely, he was defending his mother, who was accused of black magic and was in danger of being burnt on a stake... Ironic, how Protestants like to confess through projection... (Émile Namer, The Galileo Affair). Not to mention the fact, that (especially) Lutherans really detested Copernicanism, so when Kepler publicly supported it, he was excommunicated. Which was a real kick in the nutsack - it made him basically persona-non-grata in most of the Protestant world, lost him his job and almost all other income. All at the same time Galileo was just told to shut up...

That way we can state with reasonable degree of certainty, that Inquisition was not only not interested in his scientific work per se, it actually did view it as a mitigating circumstance - science is often making progress through errors. That is why, when Galileo officially withdrawn his "proof", he was let go and he continued his work more or less unmolested.

What Giordano Bruno was guilty of was a multitude of sins like heresy (repeated and quite often offense), scandal (it sounds innocuous, but in CC doctrine it is very, very bad), generally anti-doctrinal, anti-church and particularly anti-papal sentiments voiced loudly and forcefully. (Joel Shackelford, "Myth 7 That Giordano Bruno was the first martyr of modern science")

However, one thing needs clarification in this whole affair of persecution of science. If it involved Inquisition - the actual Sanctum Officium and it's officials, but NOT bodies like Spanish Inquisition - then by definition the case was doctrinal, and not secular crime. That is why Galileo was requested to show for trial by them... If he refrained from arguing his case from Biblical base, no one from the Church would give a two Messerschmits.

Bruno's case was also handled by the Inquisition with care as to the actual crimes, that is why the investigation took 8 years. The main problem was the fact that Bruno didn't know when to shut up and made a mistake of taking a jealous and vindictive man for patron, landing him in Rome's prison after nearly 20 years of speaking about his theories - mostly out of the Church's reach. Had he left the Church, Dominican Order and his vows, nothing would happen. Church, therefore, could not let it go, and he was condemned by Inquisition as a heretic. Again, normally it would not be a stake issue, but for him being officially part of the clergy, speaking agains the authority of the Church, making it political and public issue. That is why it is more accurate to say he was executed for treason, rather than for free thinking and free speaking. It is true, his multi-world theory is there, i. the trial documentation (whatever little was left), but there is no word about them in the sentencing.

So, since he was a heretic and a traitor, he was executed. It is tragic, but it must be viewed in proper context of that era. And as such, it was nothing special - there were a lot of people executed for treason around that time... Charles I of England, just off the top of my head.

Though I am sure that, for example, if tomorrow someone would publish evidence that Donald Trump was a paid spy for Russia or China, there would be quite a lot of people happy to flip the switch for electric chair he would be sentenced to. Same difference.