Not when loaded to industry specs. It IS blackpowder pressure levels.
I have said this a million times. Even if a Smokeless cartridge is loaded to the same pressure level as Black Powder, the pressure spike is very different. Most Smokeless powders have a very quick pressure spike, vs Black Powder which has a gentler pressure spike. An old Black Powder cylinder can fail if subjected to the much more rapid pressure spike of Smokeless Powder.
I have several antique Smith and Wesson revolvers and I never shoot them with Smokeless ammo, only my own Black Powder loads.
The cylinder of this Black Powder era Merwin Hulbert revolver experienced a catastrophic failure from a very light Smokeless Powder 44 Russian cartridge.
Let's look at the history of Colt metallurgy for a moment. Starting in 1873, frames and cylinders of the Colt SAA were made from materials closely resembling malleable iron, not steel. In mid 1883 frames and cylinders began to be made from transitional materials, similar to modern low carbon steel. In 1898 Colt began using low-medium carbon steels. By 1900 improved heat treating techniques allowed Colt to factory warranty the SAA for Smokeless Powder. This information comes from Jerry Khunhausen's
The Colt Single Action Revolvers. A Shop Manual, Volumes 1&2, the most authoritative book about Colts on the market today.
Back to the original poster: Colt factory warrantied the Single Action Army for Smokeless powder in 1900. Shortly after that Colt began stamping VP in an inverted triangle on the left front of the trigger guard. VP stands for Verified Proof. So if the Original Poster inspects the trigger guard, and finds the VP stamp there is no question the gun was manufactured after 1900 and was safe for standard factory Smokeless ammunition at that time.
If the Original Poster posts the Serial Number here I can look up exactly what year that Colt was made. Be sure to include any letter prefixes or suffixes, they will help zero in on when it was made. If you would like, substitute Xs for the last two numerical digits, but be sure to have the correct number of digits along with any letter prefixes or suffixes.
To be clear, the pressure vessel of a revolver is the cylinder, not the frame. To further clarify, the first Colts that shipped with 38 Special cylinders were in 1930, well into the post-Black Powder era.
So it is very likely that no matter how old the frame, the cylinder will be perfectly fine for Modern SAAMI Spec 38 Special ammunition.