Not quite.
Carrots came and come in many different colours. The most common ones were white, yellow and purple. Orange carrots did then become extremely popular and pretty much pushed the other coloured ones aside, to the point that a lot of people don’t even know carrots also come in not-orange.
It is also true that the Dutch royal family’s colour is orange and that it’s the Netherlands national colour because of that. William of Orange did indeed lead the Dutch revolt against the Spanish in the sixteenth century, he was very popular and was and still is seen as a national hero, who, in a way, was the founding father of the Netherlands.
But orange carrots were already being cultivated in Medieval Spain since the fourteenth century, perhaps by using carrots or their seeds brought to that region by traders bringing them from Arab regions using the Silk Road. We know orange carrots existed at least as far back as the fifth century AD, when a helpful artist drew a lovely and very orange carrot in the Juliana Anicia Codex, this may be a copy of an earlier book but that one sadly didn't survive.
So, the Dutch definitely did not make the carrot orange but they did make it more popular – a bit like Coca-Cola did with Santa. It was the Dutch who, when they got their hands on orange-ish carrots, went wild with them.
The little country was an agricultural and trading powerhouse at that time and they managed to tweak, grow, develop, stabilise and trade the already existing very orange carrots all over the world in huge quantities, which resulted in them becoming very common and more popular. Orange carrots went from something that existed but were not very common, to the cool, must-have carrot of the day. It wasn’t just down to their colour but because they were also a bit sweeter and ‘fleshier’ than the alternatives. Orange carrots also don’t dye food like purple ones do, making your pottage, soup or stew look a bit mucky brownish, which was probably another reason people preferred the new, hip orange carrot. They also were very easy to grow in the Western European climate.
So, we can credit the Dutch with developing and stabilising the already existing orange carrot further and making it more popular but they didn’t ‘invent’ it and it didn’t have anything to do with The House of Orange, originally. The carrot probably got its connection with the royal family much later. During politically unstable times the colour orange was sometimes banned from public view, even carrots, which of course made them only more popular among the supporters of the royals.