Why doesn't the Catholic Church kill heretics anymore?

It appears that this change was not originally on the part of the Church, but on the part of the governments on whom the Church had to rely to carry out the secular sentences (the executions). As I recall, it wasn't until this century that the Catholic Church stopped saying that the governments of historically Christian nations had a responsibility to hunt down and punish heretics in their nations; but governments had stopped supporting them long before.

The short answer, then, is this: the Church stopped sanctioning these punishments because it knew they weren't going to happen.

Note, now: the Church wasn't the party responsible for the first of these laws, and not every official or teacher in the Church agreed with them at first. Christianity itself, from the late first century to the early fourth, was not the most legal of organizations; and people were executed for belonging to it, not for falling away from it.

Laws imposing penalties on those convicted of heresy by Church tribunals were often created not by the Church's direct or indirect pressure on government, but by powerful rulers who had very strict personal interpretations of Christianity. The Roman emperor Theodosius I, with his two co-emperors, passed the first anti-heresy laws (apparently on his own and without any sort of political pressure). The 1911 Encyclopedia Britannica discusses these:

The fifteen penal laws which this emperor issued in as many years deprived them of all right to the exercise of their religion, "excluded them from all civil offices, and threatened them with fines, confiscation, banishment and even in some cases with death." In 385 Maximus, his rival and colleague, caused seven heretics to be put to death at Treves (Trier). Many bishops approved the act, but Ambrose of Milan and Martin of Tours condemned it.

In particular, Bishop Priscillian of Avila had been engaging in practices and teaching things which a synod of local Hispanic bishops declared heretical. When Priscillian appealed to Emperor Maximus to regain his status (perhaps by putting pressure on the Pope?), Maximus had him and six companions executed on charges of sorcery. Priscillian's accusers were in turn excommunicated not only by Ambrose of Milan, but by the Pope as well. Ambrose and Pope Siricius did believe Priscillian to be a heretic, but also believed that capital punishment was somewhere between unnecessary and outright evil.

Similarly, Henry V of England was a very firm Catholic who in 1414 enacted a law against heresy—under which he executed one of his old friends. The law (and two predecessors, enacted by Richard II in 1382 and by Henry IV in 1401) was primarily designed to wipe out Lollardy, which was seen as a threat to public order in light of its association with the Peasants' Rebellion (which was at most vaguely related to Lollardy). All three laws were repealed by 1553, but re-enacted in 1554 under Mary, Queen of Scots, who used them to try to re-establish Catholicism in Britain. They were re-repealed after her death.

Admittedly not all laws against heretics were of this sort; still, it does appear that (especially several hundred years ago) rulers were not only more powerful as individuals than we're accustomed to seeing today, but also more apt to put their personal religious views into law. The Church was happy to take advantage of this; Thomas Aquinas, for example, fully agreed that heretics ought to be put to death (after, as he carefully pointed out, being allowed multiple opportunities to recant):

With regard to heretics two points must be observed: one, on their own side; the other, on the side of the Church. On their own side there is the sin, whereby they deserve not only to be separated from the Church by excommunication, but also to be severed from the world by death. ... On the part of the Church, however, there is mercy which looks to the conversion of the wanderer, wherefore she condemns not at once, but "after the first and second admonition," as the Apostle directs: after that, if he is yet stubborn, the Church no longer hoping for his conversion, looks to the salvation of others, by excommunicating him and separating him from the Church, and furthermore delivers him to the secular tribunal to be exterminated thereby from the world by death.

After the Reformation, and particularly after the European Wars of Religion, governments seem to have been less likely to want to cooperate with ecclesiastical authorities in this way; and eventually the Church seems to have stopped expecting it. The Catholic Encyclopedia, written in 1909, states:

The present-day legislation against heresy has lost nothing of its ancient severity; but the penalties on heretics are now only of the spiritual order; all the punishments which require the intervention of the secular arm have fallen into abeyance. Even in countries where the cleavage between the spiritual and secular powers does not amount to hostility or complete severance, the death penalty, confiscation of goods, imprisonment, etc., are no longer inflicted on heretics.

The Catholic church stopped sanctioning these executions when State laws stopped sanctioning it.

It is similar to how Jewish high priests stopped handing over Christian heretics to the state when the Romans made it illegal to persecute Christians.

The influence of the Catholic church over the state declined starting with the French Revolution. In 1798 with French General Berthier marched to Rome unopposed, and Pope Pius VI was exiled and died shortly in 1799. His successor Pius VII continued in difficult relationship with Napoleon, "France occupied and annexed the Papal States in 1809 and took Pius as their prisoner, exiling him to Savona. This exile ended only when Pius VII signed the Concordat of Fontainebleau in 1813 (Wikipedia)".

The papal states, considered as a manifestation of the temporal power of the pope, continued in decline since Napolean. "After 1861 the Papal States, reduced to Lazio, continued to exist until 1870. Between 1870 and 1929 the Pope had no physical territory at all. Eventually Italian fascist leader Benito Mussolini solved the crisis between modern Italy and the Vatican, and in 1929 the Vatican City State was granted sovereignty. (Wikipedia)."

To some of the other answers: while there were good Catholic Christians, the atrocities committed at the top cannot be explained away or justified in anyway, even if other religions did it too.

As I said in the answer to the question you have quoted, Catholic Church did not execute people. It was the secular authorities, who were afraid that heretics would be cause of an insurrection, executed them. The main aim of the inquisition was not to put people to death but to save them from their executioners.

It was the secular authorities who stopped executing heretics when the separation of Church and state happened. When did separation of Church and state happen? That is very difficult to say as it varies geographically.

So in summary they did not stop because they never started. No formal reason was given because it was not necessary.

That said, the role of individual Catholics in burning heretics varies and this secular law has been misused by many (sometimes even by the hierarchy), but it was never done as the official act of the Church but as a secular act under the disguise of "for the good of the King and/or the country".

"Why doesn't the Catholic Church kill heretics anymore?" understood as "Why are convicted heretics no longer executed?"

The answer appears to be in this article Heresy | New Advent.

In the section Church legislation on heresy, the laws against heretics became more and more rigorous when Constantine had taken upon himself the office of lay bishop, episcopus externus, and put the secular arm at the service of the Church.

Under the purely ecclesiastical discipline no temporal punishment could be inflicted on the obstinate heretic, except the damage which might arise to his personal dignity through being deprived of all intercourse with his former brethren.

• The burning of heretics was first decreed in the eleventh century.
• The Synod of Verona (1184) imposed on bishops the duty to search out the heretics in their dioceses and to hand them over to the secular power.
• Paul III (1542) established, and Sixtus V organized, the Roman Congregation of the Inquisition, or Holy Office, a regular court of justice for dealing with heresy and heretics.
• The Congregation of the Index, instituted by St. Pius V, has for its province the care of faith and morals in literature; it proceeds against printed matter very much as the Holy Office proceeds against persons.
• In 1909, Pius X, decreed the establishment in every diocese of a board of censors and of a vigilance committee whose functions are to find out and report on writings and persons tainted with the heresy of Modernism (Encyclical "Pascendi", 8 Sept., 1907).
• The present-day legislation against heresy has lost nothing of its ancient severity; but the penalties on heretics are now only of the spiritual order; all the punishments which require the intervention of the secular arm have fallen into abeyance. Even in countries where the cleavage between the spiritual and secular powers does not amount to hostility or complete severance, the death penalty, confiscation of goods, imprisonment, etc., are no longer inflicted on heretics.

The answer

The killing of heretics stopped when the secular arm no longer intervened in penalties due heresy as deemed by the Church. (History may provide the time when separation occurred, and this may vary from state to state.)

To the question, "When did the Church stop executing people" I answer that is a false premise. Probably akin to "when did you stop beating your wife?"

To the question, "Can a reasonable differentiation be made between the actions of the state acting in the name of religion and the policies of the religion proper?" I suggest that you look into emperor Otto and how (despite his apparent contrition, immediately drove the Pope into exile as soon as his excommunication was lifted). Or, perhaps more on topic, you could look at the consistent efforts of the Papacy to stem the violence brought by the Spanish Crown during the Spanish Inquisition. Several examples:

• Near the outset of the Inquisition, in a letter of April 14, 1482, Pope Sixtus IV instructed the Spanish to ensure due process, allow legal counsel and appeal to Rome.[53] King Ferdinand defiantly rejected Papal control, the Inquisition becoming thereafter a tool of the monarchy, rather than the church.
• The Pope tried to allow for appeals, but the crown only allowed that by royal approval
• Torquemada actually systematized what could only be considered mob rule.
• And finally an appropriate quote, "In 1498 the pope was still trying to...gain acceptance for his own attitude towards the New Christians, which was generally more moderate than that of the Inquisition and the local rulers."

To the question, "When did the policy of Church enacted executions end?" This requires proof that the Church had a policy of executing criminals, as an entity independent of the state. The evidence to this is very lacking.

To the question, "When did the policy of Church-sanctioned executions end?" The answer is that it never has: given appropriate circumstances (rare in first-world countries), the Church does make allowance for the death penalty.

To the question, "When did the policy of Church-sanctioned execution of religious or political dissidents end?" Again, the answer is that it didn't. Believe it or not, the Church is not in the business of dictating punishments for crimes. I don't know of any document which has ever said that the state does not have the right to execute criminals.

I would imagine that a significant number of churchmen, and indeed a vast majority of the populous, prior to the mid-twentieth century would have allowed for executing someone who has committed treason. I would also imagine that preaching dissidence in Medieval Europe would have been viewed as treason. (This is not a real stretch, I can think of one case where speaking out against the state was punishable by torture and death in sixteenth century Holland.)

To the question, "When did the Church stop having the policy of rendering correct judgements on matters heretical?" Again, the answer is never.

To the question, "Should the Church, knowing the intentions of the state, render a correct verdict?" And I think that answer is very loaded and beyond the scope of your question.

On an interesting side note: it should be stated that the purpose of the inquisition, was not to kill heretics, but to save souls. Thomas Madden states the position:

[t]he Inquisition was not born out of desire to crush diversity or oppress people; it was rather an attempt to stop unjust executions... The simple fact is that the medieval Inquisition saved uncounted thousands of innocent (and even not-so-innocent) people who would otherwise have been roasted by secular lords or mob rule.

The article goes on to reinforce this idea by pointing out:

Madden argues that while medieval secular leaders were trying to safeguard their kingdoms, the Church was trying to save souls. The Inquisition provided a means for heretics to escape death and return to the community.

This is remarkably far from the implicit accusations in the question. I'd dare say, this was the opposite.