Yes and No...
A long accepted belief is a the 'bigger' base of a bullet allowed more pressure (from the expanding gas) to be applied AND therefore the 'wider' projectile would go faster.
What I believe to be the current accepted belief of the same phenomenon is that of 'expansion ratio' - in that the volume of the expanded gases in a a larger diameter bore arm is greater, giving more thrust to the projectile.
However, this only applies when comparing two (or more) cartridges of the same case volume and bullet weight. One example which comes to mind is the comparison between the .30 Winchester Centerfire (usually called ".30-30") and the .32 Winchester Special cartridges. Both rounds use the same basic case, the same rifle (barrel lengths and maximum chamber pressure) and much the same bullet weights. However, the .30 caliber round is .308" and the .32 Winchester is .321".
With the same powder charge and the same bullet weight (commonly they both could be loaded with 150 and 170 grain bullets, the .32 was somewhat faster from the muzzle, but the .32 projectile had less sectional density and less ballistic coefficient; therefore the smaller bullet had greater range.
The only modern (all smokeless powder and high pressure loadings) I can think of currently would be the .300 and .375 H&H Magnums, except I'm not sure what weight bullet could be used in both. (Perhaps a 180 or 200 grain bullet is available for both calibers? I just don't see a 100 to 150 grain .375" bullet or a 300 grain .308" bullet as commercially available.) Having neither caliber, nor expecting to own such, I am not at liberty to do any testing.
Some enterprising soul might try a comparison of 'identical' barrels of calibers 9x19 and .30 Luger (one could find 100 grain bullets for both, I expect) in a Contender type frame. Again, I decline on grounds of just not interested enough.
In all, the advantages and disadvantages of 'larger' or 'smaller' bore diameters (with similar bullet weights) are discussable at length. In the past, calibers .303 British, 7.92mm Mauser, .30-40 Krag, .30-06 Springfield and 7.62x54R were similar enough to inspire arguments about which was 'best' for a specific purpose.
In a purely ballistic discussion, the science is well settled. In reality, no one shoots a purely ballistic rifle or handgun. Just remember to compare any fruit with another of the same fruit.
(By the way, in the argument of 'apples or oranges', I vastly prefer licorice.)