The Twitter user concerned is being roundly condemned on Twitter and has very few defenders there. Those condemning him say that he regularly accuses people of racism and makes racist tweets himself aimed at white people and Muslims. I haven't read his tweets myself so I can't comment but I note that he referred Stewart's tweets to an organisation that combats anti-Muslim prejudice.
It seems the full quote Stewart used was:
But man, proud man,
Dress'd in a little brief authority,
Most ignorant of what he's most assur'd—
His glassy essence—like an angry ape
Plays such fantastic tricks before high heaven
As makes the angels weep; who, with our spleens,
Would all themselves laugh mortal.
The guy who complained cut off the quote after the first four lines and ignored the "man, proud man" bit to concentrate on the "angry ape". It seems they were having an argument about who pays for maintaining the royal palaces. I haven't seen the whole thread but someone else described this person as "indignantly misinformed" over his views on this subject.
if he hadn't replied with the angel emoji when someone pointed out that it was a potentially offensive thing to say
The tweet to which you are referring said, "Careful! You will be accused by the poorly educated "Shakespeare is irrelevant these days" brigade of being racist". That reads to me as a lighthearted comment. I do not see Stewart's response as doubling down on racism.
Personally, on the evidence I can see, I don't think this has "exposed" Alistair Stewart as racist as some on this thread claim.