Denial of a claim used as proof the claim is correct

I can see at least two possible fallacies here:

dicto simpliciter (accident fallacy)

"Some white people are racist; therefore, all white people are racist" is a grotesque stereotype, a sweeping generalization applied to all people of a certain race.

shifting of the burden of proof

The claim "you are a racist" would obviously require justification, so Person A in your dialogue is cleverly making Person B argue in the negative and prove he/she ISN'T a racist. That's shifting the burden of proof. As I understand it, these theories (anti-racism is another) are built on this fallacy because white people must prove they are becoming less racist  ("not racist" or "no longer racist" aren't options) to the people who created and/or teach these theories.

It has aspects of the conspiracy theory in that denials of the accusation are seen as evidence for the accusation. Mostly, it is just rhetoric.

We can also call it a simple non sequitur if the conclusion is stated clearly (or even implied).

"I'm not racist."
"That's exactly what a racist would say. So you are racist."

"I'm not an idiot."
"Idiots are too stupid to know they're idiots, so you are an idiot."

Sounds like person A is just giving an opinion. The only fact I See is "Person A: I’ve been reading “White Fragility” and I like it."

This could be multiple specific fallacies depending on how the conclusion is reached.

1) affirming the consequent

P) A racist would never admit that they are racist (because of their 'white fragility')

P) You refused to admit that you are racist (because of your 'white fragility')

C) You are racist (because you refused to admit it).

2) bulverism 

P) You denied that you were racist.

P) Suffering from white fragility means you deny that you are racist.

Implicit P) White fragility makes your denial invalid

C) Your denial is invalid, and you are racist.

3) circular reasoning 

P) You are racist because you have white fragility.

C) You have white fragility because you are racist.

4) fallacy of opposition 

P) Those who disagree with me have white fragility.

P) White fragility means you are racist.

P) If you are racist, you are wrong.

C) Those who disagree with me are wrong (and racist).

5) unfalsifiability 

Since the only way to get out of the argument is to accept its premises, there is no way to test the claims it makes (because doing so "proves" they are right, as the person is claimed to have 'fragility', and a form of fragility is expressing skepticism towards - or testing - said claims). We have an unfalsifiable hypothesis.

Unfalsifiability only means it's unscientific; not that it is false. But without good evidence, we should not accept the claim.

The fundamental problem is that the conclusion does not follow (non sequitur). Depending on context and phrasing, this could fall under many fallacies.