Well, that depends on *which* anti-Bolshevik force defeats the Bolsheviks. Geoffrey Swain in his *The Origins of the Russian Civil War* (1996) claims that "Kolchak's actions ended a war which the moderate socialists might have won and started a war the Whites would inevitably lose, putting the real civil war, the forgotten first civil war, on ice until 1920. By the time fighting resumed in Kronstadt and Tambov, the majority of Russians, after seven years of war, were no longer prepared to take up arms." p. 8 In defense of his argument that the moderate socialists might have won, he admits that the capture of Kazan in September and Samara in October certainly put the People's Army on the defensive, "but as the SRs constantly stressed during the last days of the directory, after an initial rout, volunteer units were staging a successful counter-offensive by the first fortnight in November 1918. On 5 November 1918, an offensive aimed at recapturing Samara was begun, and on 12 November 1918 the SR administration in Ufa could boast that a whole Bolshevik regiment had been taken prisoner. The successful recapture of Samara was expected with some confidence." p. 252 Personally, I think Swain lets his sympathy for the moderate socialists (which I largely share--though recognizing they messed things up pretty badly between February and October 1917) lead him to overestimate their prospects. Even if they did recapture Samara, that would hardly guarantee victory. Also, his recommendation on how the democrats could avoid a Kolchak coup--he thinks they should have taken a leaf from the Bolsheviks' book and used political commissars to assure that the army remained committed to the Constituent Assembly--might simply have provoked an earlier coup. And anyway, there was a widespread and justifiable belief that politicization of the army in 1917 had been one of the chief reasons things went wrong.
As for the Whites as usually understood--Denikin, Kolchak, etc.--I doubt they could have won (the closer Denikin got to Moscow the weaker his forces became). If they did, their official position was that they were not committing Russia to any particular form of government, whether republic or monarchy--such questions would be determined by a future freely elected Constituent 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)
http://www.rulit.net/books/a-people-s-tragedy-the-russian-revolution-1891-1924-read-232715-217.html
Genuinely free elections in Russia could simply not produce the kind of government the Whites wanted, which is why they could not in fact allow them to take place. The peasants would vote for the SR's, and the non-Russians would vote for parties that would endanger "Russia one and indivisible." Not just the Constituent Assembly elections of 1917 but the First and Second Duma elections showed the dangerous (to the Whites) radicalism of the Russian electorate--only the restrictive suffrage laws instituted by Stolypin could produce the conservative Third and Fourth Dumas.