The Huguenots, and Henri IV, are Calvinist, a doctrine that makes perfect sense to people who have freedom of travel and assembly, an economy based on trade and using cash, and at least the idea of social mobility, both upward and downward. The Huguenots are the inhabitants of cities, the professionals and wealthy commoners, and a few of the more forward-thinking nobles. It never made much progress among the peasantry (when you're on the bottom and know you're going to stay there, a Church doctrine that says low social station is proof of God's disfavor and therefore of lack of virtue isn't very appealing). At Henri IV's accession, Huguenots represented 10% of the population yet controlled 40% of the total wealth and something absurd like 80% of the liquid assets (cash) - they don't appear to have even tried to convert the poor peasantry. You can let Henri stay Protestant and get a Protestant ruling class over a Catholic peasantry, but for popular conversions you need a new man entirely - a French Luther.
Just to clarify, the bolded part was never
church doctrine. The notion that money was a sign of God's grace was a somewhat popular belief among the wealthy (naturally), particularly in the Netherlands, but was never something that Calvin preached. He maintained that it was impossible to know for certain who had received God's favor.
Also, the Huguenots certainly tried to convert as many people as they could. You have to keep in mind that Protestantism was illegal, on punishment of death, for most of the 16th century, so Huguenots weren't exactly free to preach wherever they wanted. (Calvin wisely stayed just beyond the borders of the country, in Geneva.) In areas where they had clear military control, such as Béarn and the city of La Rochelle, they converted virtually the entire population.
Part of the reason why Protestantism especially appealed to the wealthier classes was that its emphasis on reading the Bible naturally had more of an effect on the minority of the population that was literature. Also, my understanding is that church synods tended to be conducted in French (Calvin's native language), as opposed to the various regional languages such as Occitan. Outside of the Paris region, French at this time was mainly spoken by the educated classes.
As for how a Protestant France would look, I'd guess it might have its own Edict of Nantes but in reverse, with Catholics allowed certain places of worship. Or it might be like the Dutch Republic, where Catholics were discreetly tolerated. I don't think strict religious uniformity would have been enforced, at least not right away (it wasn't in OTL, even with 90 percent of the population already Catholic).