While it is unlikely to imagine the same amount of colonists coming to California from France as say, the Americans flooded the region with historically, there was historically a significant French component in the Gold Rush: up to 1/6 of the 49er's were French to my recollection. A gold rush will happen eventually and if the French themselves own the territory it is probable that they will constitute an even larger percentage, while the geography much better matches settlement there than in say, Algeria or New Caledonia, the main other French settlement regions.
There was historically a decent amount of
French interest in taking California so it isn't unreasonable to imagine.
I would suggest that California would end up substantially more ethnically diverse, probably both in terms of European immigrants (with the normal basket of Latin immigrants that the French brought along with them, plus a diverse cross section of European immigrants), and also in Asian immigrants. However, the population as a whole will be smaller. This might have some positive impacts on the Indian population at least, who will probably keep greater autonomy for longer, although just like with the Americans they will be ruthlessly squashed whenever the question of gold comes up. Regardless, it will also fit very well into the theme of the old French North America, certain to be extensively present in French propaganda, particularly since some French Canadian trappers did reach California. The convicts routed to New California historically will probably be sent to California instead. There would be very substantial investment into agriculture, and probably a fast pick up on wine and specialty crops. I doubt that, extending this to the modern day, that California will ever become the American high technology, economic center of Western North America: It will probably look more like Australia, massive, relatively low density, rich from lots of resources, but mostly focused on extraction and agriculture. With some benefits about shipping products between Asia and America.
It also seems likely that the French will pursue the Panama Canal with greater vigor, although whether they will also pursue it with greater intelligence is up for question: their original plan was completely unworkable after all. There was substantial interest in France at the time about propping up the Latin American states to attempt to contain the United States, which might also go along with this. This ties into the question raised above about defense: in the short term, given the more powerful French navy, the terrible logistics of trying to march overland to California, and the ability of the French to pour in troops by sea, they have nothing to fear from the Americans. In the long run, this will change, with greater American population and industrial power, and the development of railroads. The principal question is whether the Americans will still desire however, to fight France for California: Mexico was an easy foe and more ideologically justifiable than war against France. But it could go either way in any case.
I don't see any reason to suppose that the Americans would automatically be animated with a desire to crush France over California, after all not getting the northern part of the Oregon territory was accepted and no one today in the US cares that Vancouver is Canadian and not American. I am sure there is an alternate universe where we annexed Cuba, as often almost happened, and there are threads there about Cuba being independent, and plenty of responses about how the Americans would never stand for it and would invade the island. Hard to be revanchist about a place you never held in the first place, except from the standpoint of another universe.
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