Priti Patel said: "I have clearly upset people in the past and on reflection - and I have had time to reflect upon this as well - looking at what has been published today on the report, there is no question I'm absolutely sorry for the upset that has been caused and I'm very, very clear about that."
The prime minister said he had "full confidence" in her, despite the report author resigning over the decision to keep her in post.
Sir Alex Allan, Mr Johnson's adviser on ministerial standards, quit after saying in his report that instances of her past behaviour in the Home Office “would meet the definition of bullying” but could have been “unintentional”.
Normally ministers are expected to resign if they breach the code.
Ms Patel told the BBC: "I'm here to give an unreserved apology today and I am sorry if I have upset people in any way whatsoever.
"That was completely unintentional and I'll be very candid, you know the work that I do here in this department and across government is deeply challenging.
"So, if I have upset people, that has been completely unintentional, that was not my intention and I'm absolutely sorry for any upset that has been associated with that."
There were "no excuses", she said, and that she was giving an "unreserved, fulsome apology".
"I cannot be any clearer about that whatsoever," she added.
The partially published report, which came during national anti-bullying week, said the home secretary had not consistently met the standards required by the ministerial code.
Earlier, Ms Patel admitted becoming “frustrated” in dealing with colleagues. In a statement she said: "I am sorry that my behaviour in the past has upset people.
"I am very grateful for the hard work of thousands of civil servants who help to deliver the government's agenda.
"I care deeply about delivering on the commitments we have made to the people of this country and I acknowledge that I am direct and have at times got frustrated.”
Allegations of bullying have been made against Ms Patel for many months. Questions were raised about her conduct when in February the Home Office's most senior civil servant, Sir Philip Rutnam, resigned amid reports of friction between the pair.
Sir Philip said other people had repeatedly accused the home secretary of bullying, and he argued her behaviour had "created fear".
Ms Patel's behaviour while she was international development secretary also came under scrutiny, and the inquiry was set up in March.
Sir Alex’s report said Ms Patel was unaware of the impact of her behaviour because no feedback was given to her at the time.
Mr Johnson's official spokesman said: "The prime minister took advice from his independent adviser Sir Alex Allan and, as you've also seen, the home secretary has this morning apologised.
"The PM is reassured that the home secretary is sorry for inadvertently upsetting those whom she was working with but, ultimately, as the arbiter of the ministerial code, weighing up all of the factors, the prime minister doesn't believe there is a breach.”
He added Mr Johnson was reassured that relationships in the Home Office are much improved.
The government said Mr Johnson "considers this matter now closed".
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies