In the Middle Ages, it was necessary for a couple to have as many children as possible because many children died before reaching adult age. Much of this was due to rampant disease and lack of medical knowledge. In an alternate history, diseases are rare and easily curable through the use of magical herbs. Most children born survive to adulthood. Women dying in childbirth is extremely rare. Would this cause overpopulation during this time period?
There is still starvation
I would argue that starvation and susceptibility to disease are intertwined. The people who did the most dying of diseases were the ones who were starving. It is worth noting that European population exploded in the 1700s to 1800s; before modern and universal medicine, and before widespread immunizations. However, this was after an Agricultural revolution increased the farming yield, and after improved transportation technologies allowed food to be moved from distant places to combat local famine.
So I assert that starvation is more important than disease as a means of population control. A lot of medieval disease deaths simply took people that would have starved anyways. But even with no diseases, people aren't going to make it if there isn't enough food.
With few children dying due to disease and women generally surviving childbirth, I don't see the same need for the "have as many children as possible" mindset.
That said, farmers and a few other jobs would probably still want to have as many "free labor" as possible. Having more people farming would have helped feed the not diseased masses.
Not having to fight disease as much would have freed people up to make more discoveries, like better farming and industrial practices.
Without the pressing need for most people to have a large family, I don't see as great a possibility for overpopulation during that time frame. There would definitely be more people around, but without calculating the food supply available, we can't know if this would be "over population" as we think of it.
You also have to remember that wars were sometimes fought as a way to control over population. I have a suspicion that there would be more wars fought, due to this.
However, in the long term, since families didn't have the long standing need to procreate against loss, we currently wouldn't be facing the large crowds of people we have today. Nor would we see that ever expanding population in the future grow as quickly as we do.
If you have magical medicine, then you better get a magical Norman Borlaug. Otherwise, you will have overpopulation and famine.
If the same handwaving that solved disease can be used to boost agriculture, you basically have enough population to do anything you want, relative to other countries.
It might not. One good example are Tibetan and other Himalayan societies, where their environment supports so few people that even small population growth is not very welcome. They came up with some cultural solutions that could easily be adapted to a medieval European setting:
It's hard to find exact numbers on this, but apparently in some areas at some times, about one in three men was a monk, and thus did not marry. This may be a bit misleading because Buddhist monks aren't necessarily expected to stay a monk all their life, but most figures I can find posit at least 10% of Tibetan men living in monasteries (before the 20th century). Sending surplus sons to the monastery was also a common way of managing your inheritance in European history.
Polyandry, where one woman can have multiple husbands, is quite common in Himalayan societies and really rare in the rest of the world. The men will obviously have fewer children than if each man had his own wife, so this limits population growth. Alternatively, you could have some form of extreme polygyny, where high-status men have a lot of wives, and most lower status men don't get to have a wife at all. But I feel like that society would be quite different from medieval Europe, and I don't think any agrarian societies worldwide work like that.
While having a lot of children to do labour is an advantage in agriculture, you do have to worry about what will become of them once they're all grown up. In a society with low mortality, there will probably be a lot of adults around who don't have their own land (because a sibling inherited it, or because it was too small to live off so they sold it, or because their parents didn't have land either). These people would be a good source of labour and probably not much more expensive than a child -- and certainly more productive! So your society, having access to pretty effective medicines, will probably work out a way to limit reproduction for married women. In some Himalayan societies, herbal abortions were quite common, and young people were taught how to have non-reproductive sex.
If your society does not adapt to its circumstances at all, and stubbornly maintains that everybody needs to have as many children as they can, they will certainly collapse. But I feel that issues of inheritance and wealth distribution will lead to societal adaptations before starvation does.
I'd like to offer an alternative point of view to the one provided above:
Considering the definition of overpopulation:
the condition of having a population so dense as to cause environmental deterioration, an impaired quality of life, or a population crash
one could say that europe was already overpopulated. The reasons may have been partly artificial in the form of wars, feudalism and serfdom, but the general population was at the maximum that was bearable by the system. So even if the medical aspect would have been taken out of the equation, famines and war would have kept population approximately on the same level or at best slightly above. People would have most likely adapted with a lower birth-rate to the fact that they rarely died of diseases.
So it would not have significantly changed the situation. There was already overpopulation.
Yes it can. However it most likely will not!
If the lack of diseases was a constant on your world then the populations evolved with this and will lack the need to have a large number of children in fact more children will be an hindrance on the family and society will become one that discourages having more children.
However lets assume that this is an recent event! And the population has just started spiking.
When that happens then the villages and cities will require more food! They will either accept their fate and die or wars will break out. There is also a change that populations might start migrating to underpopulated areas or virgin lands.
I see no way that you might have overpopulation in this world.
If nearly no kid dies because of diseases (even if they are weak beacause of malnutrition) the parents would naturally get a lot less children than you normally got during the real medieval times. If they are sure to keep all their children they could even plan how many they needed to make the job at the farm easier.