Trump spent five years trying to delegitimize Obama. Now he's taking office under a cloud of suspicion, and only has himself to blame. It is ironic that Donald Trump owes his success in Republican politics, and thus ultimately his presidency, to birtherism given the extent to which he failed to bring birtherism mainstream.
Donald Trump is no different than anybody else out there. He is a media whore who thrives on attention by spouting off ridiculous lies and rumors that prey on people's paranoia and fears. His cloud of suspicion and people closely questioning his legitimacy as the new president of the United States is his own actions coming back to him. When you enter into a position of power like this you need to be prepared for all of your dirty laundry to be aired.
Trump entering office under a suspicious cloud is unavoidable. Backing off on his racist and unnecessary demands of seeing Obama's birth certificate would not help him in any way. Trump is a racist, homophobic, misongist and unfit political candidate. His career is filled to the brim with scandals and he should be looked at under a powerful microscope as he takes on the position of President.
Without the dirty tactics Trump used, such as questioning Obama's birth certificate, he would never have got the support he needed. Those sort of things attracted certain type of people that would vote for him. That means laying off those tactics back then wouldn't have got him in to this position. Although it means less support now, he had to do it in the first place because he was never going to win based on actually being qualified for presidency.