The answer clearly states that there is no rejection.
There is nothing in the sentence that hints to rejection.

Only a confession, and the following embarrassment

It's based on your sentence at the end of the question. where you write:

"Did she just reject him?", to which my answer is: "No, she did not just reject him"

To sketch you a similar, but different scenario.

Imagine during your school days, say at around the age of 14-15.
You are in class, and accidentally call your teacher mom.
What would have been the reaction you'd have at that moment?

Could you imagine how you'd respond in such a scenario? (You don't nececarly have to describe it)

But most likely, you would feel somewhat embarresed.

The scene in the anime envisions something similar.
All though instead of calling a teacher mom. It would be more like asking for a teachers hand in marriage, in front of the whole classroom by accident.

In essence, what she was trying to say was something along the lines of "I would like you to know I have feelings for you" In a shy, and very formal way.

But what she actually ended up saying was something more along the lines "I would like you to know, that I would like to be with you till death do us part (wedding vow), I, I, thats not how I meant to say it"