King Henry II of France was the King of France from 1547 to 1559. He was the son of Francis I of France, and husband to the famous Catherine de' Medici. He continued his father's persecutions of Huguenots in France and conflicted with the Habsburgs. His reign was cut short when he got severely injured during a joust which led to his death. Afterwards, he would be succeeded by his sons (Francis II, Charle IX, and Henry III) all of whom died either young, or heirless, spelling the end of the Valois Dynasty and paved the way for the Bourbon Dynasty, while also beginning the French Wars of Religion.
So, what if Henry II of France didn't get injured in a jousting accident and lived a little longer. Without his death and the premature succession of his sons, how would this effect the French Wars of Religion? And how would it effect geopolitical aspect of Europe at the time?
Henri II was very anti-Protestant. While his wife and sons always sought conciliation, he is likely to fight and persecute Protestants as much as he can. Here, we have two possibilities.
Either the Protestants are quickly crushed and most of them flee France or convert to Catholicism out of terror. Either the Protestants manage to resist and the civil war is even more destructive than in OTL.
If he ends up outliving Francis II, would it be possible for MQOS to marry Charles IX? It might have been possible for him to have a fifth son after Anjou
I guess you mean a
sixth son. Five is the number of sons Henri II and Catherine had in OTL. Or maybe you doesn't count Louis?
In any case, as Tudorfan said, Henri II and Catherine having other children is very unlikely. Catherine had decided to not have children anymore after the twins. She is likely to refuse sex until menopause.
As for a marriage between Charles and Mary, I see it as very likely, even almost sure.
Let's notice Francis has a little chance to live longer. Just before his OTL death, Ambroise Paré had proposed a trepanation in order to try to save him. Catherine de Medici refused because she saw it as too dangerous. With Henri II alive, he would be the one to decide, not Catherine. Francis may gain some time but, considering his terrible health, I am pretty sure he would still die very young anyway. I hardly see him reaching the age of twenty.
Henri II wanted the alliance with Scotland very much and he hoped to also make Mary queen of England and build a united kingdom of France, Scotland, England and Ireland. So, yes, as soon as Francis is dead, Henri II would order a marriage between Mary and Charles. This marriage is likely to be fruitful: both Mary and Charles were fertile and, even if Charles still dies of tuberculosis at the age of twenty-three (which is far from sure), they would probably marry as soon as Charles comes to age, giving them enough time to produce children.
As a result, we would have future kings of France and Scotland and claimers to the throne of England.
Of course, Mary and Charles' descendants are unlikely to manage to get control over England. Maybe Elizabeth I would marry and try to produce children. If she does not, I guess the throne of England would pass to some descendant of Mary Tudor the Elder. Even English Catholics would probably refuse to support Mary as they would not want a personal union with France and they would have Darnley as a Catholic claimer. The French claim to the throne of England is likely to end up like the English claim to the throne fo France: something purely formal. As for Scotland, Mary and Charles' eldest son would be the undisputable heir to the throne but, in OTL, Francis and Mary had lost effective control over Scotland when he died. I am unsure how things would turn out about Scotland.
Also, Charles having a son implies his younger brother Henri never becomes king of France. Would he still become king of Poland? I think so. Henri II would probably like the idea of his younger son becoming king of another country and would support the election. If the young Henri stays in Poland, what would his reign look like? He disliked Poland in OTL but he stayed only three months. Had him stayed longer, he would've probably adapted. And, of course, it implies Stephen Bathory stays in Transylvania and Sigismund Vasas stays in Sweden, which is also very interesting (Sigismund being the Catholic king of a Protestant country).