On 13 December 2015, the web site Allen West Republic published an article about the swearing in of Judge Carolyn Walker-Diallo, titled "Obama’s America: Muslim Woman Sworn in As Judge, Look at the FIRST Thing She Does!":
New York Judge Carolyn Walker-Diallo, a Black Muslim woman took oath as a civil court judge in New York on Thursday, while swearing to abide by the U.S. Constitution placing her hand on the Holy Quran.
The article's primary assertion was that the "first thing" Walker-Diallo did upon becoming a judge was to was place her hand upon the Quran, although technically she didn't become a judge until the swearing-in was complete. (Judges are sworn in using a Bible, religious text of their choice, or a copy of the U.S. Constitution.)
A similar controversy erupted regarding Minnesota Congressman Keith Ellison's use of a Quran in 2006. At that time, UCLA law professor and legal analyst Eugene Volokh responded to a CNN panel question about whether the use of religious texts other than the Bible was addressed in law, explaining (in a response that has nothing to do with President Obama):
Well [the Constitution] actually does say a couple of things. First, it doesn't even require congressman to use any religious text or any religious component. It specifically provides that they may affirm, rather than swearing. That was for the benefit of people who have a religious objection to invoking God in an oath.
Quakers were a traditional example. And for example, President Herbert Hoover was sworn in without putting his hand on any book.
It also says no religious text shall be used for government office. And when you're required to swear on the book of a religion that is different from you, not traditionally you've done it, that would be an impermissible religious test.
As for the claim's veracity, New York State Senator Martin Malave Dilan published the following status update to Facebook on 11 December 2015:
Multiple news sources reported that Walker-Diallo used a Quran as her religious text of choice for her swearing-in (noting the controversy it engendered), and the event was captured on video:
A New York City judge is facing a backlash after taking her oath using the holy book of Islam in recognition of her Muslim faith. Carolyn Walker-Diallo was elected in Brooklyn’s 7th Municipal District and took her oath of office last week using the Quran.
After attenders posted a video of the swearing-in session to social media, conservatives unleashed a storm on insults toward the public servant. "Sickening," one Facebook user posted in response to the video. "Is this America or the Middle East." "Another piece of s--- Muslim," another user wrote, "trying to take over this country."
The President of the United States is not involved in the selection of New York City municipal judges.