Morgan left the show on Tuesday after casting doubt on Meghan Markle's comments about her mental health, with Osbourne subsequently tweeting in support of him - and her remarks prompting a backlash.
The following day Osbourne, 68, appeared on US show The Talk where she spoke about the criticism in an emotional outburst, as she said: "I feel like I’m about to be put in the electric chair because I have a friend who many people think is a racist so that makes me a racist."
In a message to Twitter on Thursday night US time, the former X Factor judge said after “some reflection”, that she is “truly sorry”.
She wrote: “I have always been embraced with so much love and support from the black community and I have deep respect and love for the black community.
“To anyone of colour that I offended and/or to anyone that feels confused or let down by what I said, I am truly sorry.
“I panicked, felt blindsided, got defensive and allowed my fear and horror of being accused of being racist take over.
“There are very few things that hurt my heart more than racism so to feel associated with that spun me fast! I am not perfect, I am still learning like the rest of us and will continue to learn, listen and do better.”
Watch: Sharon Osbourne tearfully defends Piers Morgan
Osbourne was criticised over the discussion which saw her ask co-host Sheryl Underwood, who is Black, to "educate" her on racism.
She said: "I will ask you again, Sheryl. I’ve been asking you during the break, and I’m asking you again.
“And don’t try and cry, because if anyone should be crying, it should be me.”
Osbourne continued: “This is the situation. You tell me where you have heard him say… educate me! Tell me! When you have heard him say racist things. Educate me! Tell me!”
The mother-of-three had originally said she "stood by" Morgan for "speaking his truth" on the breakfast programme.
Now she has said: “Please hear me when I say I do not condone racism, misogyny or bullying.
“I should have been more specific about that in my tweet. I will always support freedom of speech, but now I see how I unintentionally didn’t make that clear distinction.”
Morgan described Osbourne as a "true friend" when responding to her support.
Watch: Piers Morgan insists 'he won't be going back' to Good Morning Britain as petition calls for return
With additional reporting by PA.