The Whites would
have to govern dictatorially, whether they wanted to or not. Yes, they paid lip service to a Constituent Assembly. They kept on saying that everything--the status of the monarchy, the land question, etc.--would ultimately be determined by such an Assembly. (No doubt this was largely because they knew that such questions as monarchy versus republic would divide them, so talking about the Constituent Assembly was a convenient way of putting them off until the future. As Denikin wrote in 1918, "If I raise the republican flag, I lose half my volunteers, and if I raise the monarchist flag, I lose the other half. But we have to save Russia." "For this reason, the army's slogan was not any specific form of government, but 'great Russia, one and indivisible.'"
https://books.google.com/books?id=NAZm2EdxKqkC&pg=PA209)
However, whether they would really allow such an Assembly to be freely elected is doubtful. Kolchak's testimony seems to indicate the Constituent Assembly the Whites had in mind (or at least that he had in mind, but I doubt that Denikin would think differently) was not the democratic one elected in 1917 (and which was overwhelmingly dominated by self-described socialists of one sort or another, as IMO any democratically elected Constituent Assembly in Russia at the time would be):
"The general opinion...was that only a government authorized by the Constituent Assembly could be a real one; but the Constituent Assembly which we got...and which from the very beginning started in by singing the 'Internationale' under Chernov's leadership, provoked an unfriendly attitude...It was considered to have been an artificial and a partisan assembly. Such was also my opinion. I believed that even though the Bolsheviks had few worthy traits, by dispersing the Constituent Assembly they performed a service and this act should be counted to their credit." (Quoted in Orlando Figes, *A People's Tragedy: The Russian Revolution 1891-1924*, p. 588)
https://webcache.googleusercontent....n/a-peoples-tragedy-russian-revolution/41.php
Any free election would give a victory to socialist parties (and the Whites hated the moderate socialists almost as much as they did the Bolsheviks) and to non-Russian parties (also largely socialist) advocating extensive autonomy if not outright independence for the groups they represented. This was against everything the Whites believed in. (On the national question, there was one exception, Kornilov, who was even willing to accept a self-governing Ukraine,
https://books.google.com/books?id=irWQQCXwhwwC&pg=PA42 but the other Whites considered him hopelessly naïve where politics was concerned, and anyway he died early in the civil war.)
You don't even have to look to the OTL Constituent Assembly elections to show that free elections in Russia were likely to produce results unsatisfactory to the right, even the center-right. Look at the election to the Second Duma in 1907...