Some of Obama's actions as President were instrumental in ensuring that DeGeneres and her wife are considered to be married under US federal law and the laws of all 50 US states. This is a reasonable interpretation of the phrase "legally married".
DeGeneres and Portia de Rossi were married in California in August 2008. The marriage was valid under California law at the time, due to the California Supreme Court ruling in In re Marriage Cases as of May 2008. I know of no evidence that Obama had any role in that case. The marriage has been valid under California law ever since; the November 2008 passage of California Proposition 8 prohibited new same-sex marriages but did not affect the validity of those already performed.
However, the marriage was initially not recognized under US federal law, due to Section 3 of the Defense of Marriage Act (DOMA) which was in force at the time. So DeGeneres was not yet a "legally married woman" for the purposes of federal law. She also would not have been "legally married" under the laws of other states which declined to recognize California's same sex marriages, a stance which Section 2 of DOMA permitted.
Section 3 of DOMA was struck down by the US Supreme Court in U.S. v. Windsor in a 5-4 decision in June 2013. Since that ruling, DeGeneres and de Rossi's marriage has been recognized by the federal goverment. Section 2 was invalidated in Obergefell v. Hodges in June 2015, also 5-4, and since that date the marriage has been recognized by all 50 state governments. Obergefell also required all 50 states to allow same-sex marriages to be licensed and performed, which was perhaps the more prominent effect of the ruling; but this aspect was not directly relevant to DeGeneres and de Rossi, whose marriage had already been legally performed under the laws of their state.
Obama did have a role in the invalidation of DOMA.
There were some changes and inconsistencies in his public statements on same-sex marriage, but for instance, the 2008 Democratic Party Platform contained a statement opposing DOMA.
Obama appointed two Supreme Court justices (Sotomayor in 2009 and Kagan in 2010) who would go on to vote with the majority in both Windsor and Obergefell. Since both decisions were 5-4, the votes of both justices were crucial. I do not know to what extent their views on the legal status of same-sex marriage were a factor in their nomination, but it certainly would have been in Obama's power to appoint justices who were staunchly opposed; he did not do so.
Obama announced in 2011 that he believed DOMA was unconstitutional and was directing the Justice Department not to defend it in court. The case for DOMA continued to be argued by the Bipartisan Legal Advisory Group of Congress, so it is difficult to tell what effect this action had on the law's eventual fate, but certainly it withdrew legal resources that otherwise could have assisted in defending it. (Thanks to Mark for pointing this out in a comment.)
Based on this, I think it is reasonable to assign Obama some credit for the fact that DeGeneres is a "legally married woman" under federal law.
The Daily Caller's counterclaim that Obama opposed same-sex marriage until 2012 may be in reference to a 2012 statement in which he said unequivocally that "I think same-sex couples should be able to get married." Prior to this, he had not supported federal efforts to require all 50 states to perform same-sex marriages. However, as far as I know, he had not moved to prevent individual states from performing same-sex marriages if they chose to make it legal, and he had supported efforts to require the federal government and all states to recognize same-sex marriages. In my view, it is these latter issues that are most directly relevant to DeGeneres's case.
(I have used Wikipedia as a reference only for laws and court cases whose existence is not in dispute, as a source for background information and links to the text of the laws, court filings and opinions, etc. If people have other references offering similar breadth, they are welcome to edit them in.)
