Phandu Skelemani made the apology after Kenya's government accused him of being contemptuous towards Uhuru Kenyatta.
The International Criminal Court has charged Mr Kenyatta with crimes against humanity.
He denies the charges and won the 4 March poll with an outright majority.
He got 50.07% of the vote compared to his rival Raila Odinga's 43.31%.
Mr Odinga has refused to accept defeat, saying the election was marred by fraud and is expected to file a petition challenging the result at Kenya's Supreme Court on Friday.
Analysts say the ICC case bolstered Mr Kenyatta's electoral chances, as voters saw it as interference in Kenya's domestic affairs.
He is due to stand trial in July, with the charges stemming from allegations that he helped orchestrate violence that erupted after the 2007 election.
Earlier this week, his lawyer, Steven Kay, called on the ICC to drop the case, saying the evidence against him was "utterly flawed".
Mr Kay made the call after ICC chief prosecutor Fatou Bensouda dropped charges against Mr Kenyatta's co-accused, Francis Muthaura, because a key witness had "recanted" his evidence and other witnesses were too scared to testify.
Mr Skelemani said he was retracting comments he made to Botswana's privately owned Mmegi newspaper following Mr Kenyatta's election victory.
"I apologise to the Kenyan people for my earlier statement and wish to maintain that Kenya and Botswana have always worked together and nothing will change that," he said, in a statement published in the Kenyan media.
"Mr Uhuru Kenyatta is more than welcome to visit Botswana. Botswana is cognisant of a section of the law that says one is innocent until proven guilty."
Earlier this week, Mr Skelemani was quoted by Mmegi as saying that Mr Kenyatta would not be allowed to "set foot" in Botswana if he refused to cooperate with the ICC.
"If he refuses to go [to The Hague], then we have a problem. That means that they do not know the rule of law," he is quoted as saying.
"You can't establish a court and refuse to go when it calls you."
In his response, Kenyan government spokesman Muthui Kariuki told the Nairobi-based Daily Nation newspaper that Kenya knew its international obligations but would not be bullied.
"The government finds the statement contemptuous," he is quoted as saying.
"Mr Kenyatta has been accused, but he is not guilty until proven otherwise. In fact, he has always attended ICC sessions without failure and is on record he would continue to do so," Mr Kariuki said.
Botswana has also warned Sudan's President Omar al-Bashir, who has been charged by the ICC with genocide, that he will be arrested if he visits Botswana.
He has denied the charge, but refuses to stand trial because Sudan does not recognise the ICC, saying it is a tool of Western powers.
Botswana's foreign minister has apologised for saying that Kenya's newly elected president would be banned from the southern African nation if he refuses to co-operate with the ICC.