It depends on the scale. If space data centers become a thing at the scale Musk and others are talking about (millions of space data centers) then rockets will indeed became a major issue for the atmosphere, as will the space data centers themselves.
The total atmospheric pollution from rockets, even at that scale, will be small compared the total from terrestrial sources but unfortunately atmospheric pollution works like real estate: location matters.
Orbital rockets deposit stuff into the upper stratosphere which is about the worst place you can put. Things there have effects massively disproportionate to the same amount of the same things emitted into the lower atmosphere by terrestrial sources. Some have up to 500x the effect when emitted in the upper stratosphere than when emitted by airplanes or on the ground.
It's been OK so far because in the entire history of rockets there have only been about 7500 orbital launches (which are what matter for the upper stratosphere--see below). With a million space data centers which are expected to each have a useful life of 5 years there would have to be about 2000 Starship launches (@100 data centers per launch) per year just for replacing the ones that have reached EOL. That's after the 10000 launches for initial deployment.
Worse, satellites damage the upper stratosphere on reentry. With a million space data centers that's 200k reentries per year. The cumulative number of reentries so far is around 10k satellites and spent rocket bodies, and around 25k if we count smaller things that we can track like debris.
There have been tens of thousands of suborbital launches that have reached the upper stratosphere but most of them have been sounding rockets which are small enough that the cumulative pollution from all of them is less than a year's worth of current orbital launches. On top of that most of the sounding rockets were for weather monitoring and have been replaced by balloons. Current sounding rocket use is negligible. There have also been ballistic missiles but by the time those are in the upper stratosphere they are ballistic and don't pollute much. Hence it is only orbital launches we have to worry about.