They did. The treaty stated it this way:

The inhabitants of the ceded territory shall be incorporated in the Union of the United States and admitted as soon as possible according to the principles of the federal Constitution to the enjoyment of all these rights, advantages and immunities of citizens of the United States, and in the mean time they shall be maintained and protected in the free enjoyment of their liberty, property and the Religion which they profess.

Jefferson was famously concerned about the constitutionality of the executive branch acquiring territory. Very few shared this concern. There was more concern, however, about whether it was legal or ethical to annex territory without the consent of the territory’s citizens.

This issue was debated in Congress. John Quincy Adams argued that “we ought to have applied to the inhabitants of Louisiana to recognize our right to govern them.” Few other congressmen agreed with him, it turned out. And in any case the debate was more about the abstract principle than the actual case with Louisiana, since it was assumed that “the inhabitants would readily agree to the proposed transfer of their allegiance; […none could] be so unwise as to prefer being the colonists of a distant European Power, to being members of this immense Empire, with all the privileges of American citizens”, as Madison put it.

Even so, it was not clear how this “transfer of allegiance” would work. US politicians knew that Louisiana was effectively an imperial colony of foreigners. They differed over what ought to be done to “Americanize” Louisiana. Many, including Jefferson, thought it was essential to “dilute” the population with American settlers and to apply some kind of acculturation pressure.

Jefferson suggested annexing Lower Lousiana to Mississippi Territory, in order to mix its people “with such a body of Americans as may take the lead in legislation and government”.

That quickly proved impractical. Instead he divided Louisiana into two interim territories. The southern territory, which had about 50,000 inhabitants, mostly French-speaking, was made fairly large, to make it easier for American settlers to “dilute” or “Americanize” the population.

Most US leaders saw the people of Louisiana as foreigners, speaking a foreign language and adhering to a foreign culture. It was assumed that they were used to authoritative government. They followed strange legal customs and laws and would need help setting up American-style representative institutions. They would need a strong American governor and close federal supervision. Their laws, courts, etc, would have to be remade to conform with the American system. Louisiana followed the model of French and Spanish civil law, while English common law was the model in the rest of America. Americans generally assumed their system was superior and had to replace the civil law system in Louisiana.

One obvious way to do all this was to encourage American settlement in Louisiana. Jefferson supported a bill that would pay for 30,000 Americans to settle in Louisiana. They would receive 160 acres of land and the government would pay for at least their transportation to Louisiana. The bill passed in the Senate, but stalled in the House.

The people of Louisiana were, surprise surprise, not happy about these things. They had many grievances under the new territorial government. There was social unease and conflict. A delegation went to Washington to plead for immediate statehood and the preservation of at least some of Louisiana’s laws and customs. Louisianians resisted most attempts to change their civil law system. They won a major victory in 1808 when the US formally recognized Louisiana’s Digest of the Civil Laws, modeled on the Code Napoléon. The US also dropped its insistence on the use of English for all government business (the laws, for example, were printed in English and French).

One aspect of American law that Louisianians were happy to adopt was trial by jury in criminal cases.

Anyway, American feelings about this alien people they had annexed eased in the wake of the Burr Conspiracy. The French Louisianians were not at all implicated in the intrigues and suspected treasonous cabals. On the contrary, it was the allegiance of Americans in the region, not the French, that was concerning.

In the changed political climate following the Burr Conspiracy Congress was much more willing to consider statehood for Louisiana. This reopened many debates, and many northern politicians spoke out against statehood, sometimes citing the unamericanness of the “Hispano-Gallo race” (though slavery-related things were perhaps the main issues of debate). It took a few years, but Louisiana statehood was approved in 1811.

Nevertheless, Louisiana retained the feel of being an imperial colony divided between the French inhabitants and a growing number of Americans. Over time the “imperial” aspect of Louisiana faded, but its “unusual” culture survived in many ways.

To sum up, the annexation of Louisiana and its inhabitants was hugely significant. Neither America nor France had made the slightest gesture of interest in how the citizens of Louisiana felt about it. The Constitution said nothing whatever about such a case. It was, in effect, a sudden experiment in empire. Previously, Americans sometimes spoke of being an “empire” in a poetic, rhetorical sense. Now America was an empire in the real sense of having annexed peoples of foreign culture, who had not chosen to be Americans. Of course, Native Americans had long been exactly that, but had long been put aside as special case and not given citizenship. What was done to the French colonists of Louisiana was seen as something quite different and with the possibily of far-reaching consequences for the United States itself.

Sourced mainly from The Shaping of America, Volume 2, D.W. Meinig.

(edit: this word that word; tpyos)

Was this this same situation as when the US bought Alaska? Did the people already in Alaska become US citizens as well?