There are two different (but related) issues at play here to be discussed: Theodore Roosevelt's attempt to raise a division to fight overseas, and the forced retirement of Col. Charles Young. The answer to your question is racism played little if any role in the decision to deny Roosevelt's offer to lead troops, but that's primarily because circumstances didn't exactly align in a way for that to happen.

The decision from the Wilson administration not to allow TR to raise a regiment of volunteers was primarily one of military organization. TR was seen as by the Wilson administration as overly enthusiastic without the requisite skills, experience, or temperament that this conflict would require. Secretary of War Newton Baker, who was the person most responsible for this decision, Wilson taking his advice, was opposed to volunteer forces on principle, and especially the idea of one led by TR. As were many other officers and staff in the military. Baker believed fully-trained, regular army soldiers, commanded by experienced leadership, was necessary.

I could not reconcile my mind to a recommendation which deprived our soldiers of the most experienced leadership available, in deference to any mere sentimental consideration, nor could I consent to any expedition being sent until its members have been seasoned by most thorough training for the hardships which they would have to endure. I believe, too, that should any expeditionary force be sent by the United States, it should appear from every aspect of it that military considerations alone had determined its composition, and I think this appearance would be given rather by the selection of the officers from the men of the Army who have devoted their lives exclusively to the study and pursuit of military matters and have made a professional study of the recent changes in the art of war. (Letter from Newton Baker to Theodore Roosevelt, April 13, 1917.)

(For those interested, you can read the full published correspondence between Secretary of War Baker and TR here: https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o286988.)

In his book The Approaching Storm: Roosevelt, Wilson, Addams and Their Clash Over America's Future, Neil Lanctot quotes Baker and others in saying that they felt TR "utterly unqualified" to lead any kind of troops. He had served in the military essentially for a matter of weeks (months, at best), and that had been almost twenty years prior. He was also nearing 60 and not in the best of health. (Lanctot 557-567)

It might be tempting to assume Wilson and Roosevelt's bitter opposition to one another affected the decision, but that doesn't appear to be the case either. John Milton Cooper, Jr., in his book The Warrior and the Priest: Woodrow Wilson and Theodore Roosevelt, argues that in addition to the recommendation of the War Department, it was Wilson's very philosophy of the conflict that led him to reject TR's proposal. Wilson, Cooper says, "projected his belief in self-control onto the nation. Wilson dismissed his rival for stirring up rather than bridling passions and for seeking total victory instead of 'peace without victory.'" (Cooper 326)

As to the issue of Lt. Col. Charles Young, the possibility of his involvement was not disclosed to the Wilson administration. Young, who was a seasoned veteran and civil servant - an early Black graduate of West Point and the first Black man to obtain his rank - was being seriously considered for a promotion to brigadier general. However, because they could not abide by the possibility that a Black man might lead White troops, Young was retired due to supposed high blood pressure and kidney issues.

In May 1917, Young reached out to TR, asking him to be involved should he plan to recruit Black soldiers to fight in Europe. (Read the initial letter here: https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o279508. We have other correspondence from this chain as well.) As it so happened, Roosevelt had already intended to raise one or two individual regiments of Black soldiers for his division, and he offered Charles Young command of one of these. But, critical to answering your question, TR did not inform Baker or Wilson of this during their discussions.

(Curiously, TR did inform Baker of many of the officers he intended to recruit. Did he intentionally leave out Young? It is possible, but this is only speculation. Especially considering that TR began his correspondence with Young in early May, and ended it with Baker in mid-May; it may have just not been possible to mention Young or Black regiments even if he'd wanted to.)

So, although racism had certainly been involved in the decision to force Young's retirement, leading him to seek out volunteer opportunities with TR, his involvement was seemingly not known to the War Department. But as it stands, they were fully prepared to reject any involvement by TR without that knowledge anyway.