In short: separation of (coequal) powers means the President can't order any such thing of Congress. Congress does as it wills, and the constitution has very little to say about whether it does its jobs in any particular time frame, or even in any particular way.
Article 2, Section 3 of the constitution details the two things a President can force Congress to do (emphasis mine):
He shall from time to time give to the Congress Information on the
State of the Union, and recommend to their Consideration such Measures
as he shall judge necessary and expedient; he may, on extraordinary
Occasions, convene both Houses, or either of them, and in Case of
Disagreement between them, with Respect to the Time of Adjournment, he
may adjourn them to such Time as he shall think proper; he shall
receive Ambassadors and other public Ministers; he shall take Care
that the Laws be faithfully executed, and shall Commission all the
Officers of the United States.
So his only power over congress is to force them into session (in "extraordinary occasions") or to settle a disagreement on their adjournment. The last part I highlighted is also particularly relevant to your consideration. While Article 2, Section 1 does specify that he will take an oath to uphold the constitution:
Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:-"I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."
Article 2, Section 3 above specifies only that he take care to faithfully execute the laws as an actual duty.
Congress has certain constitutional powers and obligations, but it sets its own rules on how it does these things in almost every instance. Article 1, Section 5 includes the clause:
Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.
The Supreme Court has generally ruled that the rules of the houses are essentially non-justiciable: they are not subject to overview by the judicial branch. Even the constitutional requirement for a quorum in the House/Senate is in the sole purview of the rules of the chambers themselves, and the courts have no power to dictate or question when the requirement has or hasn't been met; only the houses themselves determine this, by whatever means they see fit, and if the rules say there's a quorum, then there's a quorum.
Obama, having a law degree from Harvard, and professional experience in constitutional law specifically (he was a professor thereof for a time), was certainly aware of all of that. At best he could have tried to exert political pressure on Congressional Republicans, say by being more forceful about the matter in public statements. Some people do feel he didn't do enough of this, some even feeling he should have done more even if it was guaranteed to not change the outcome. But that's more of a political contention than a constitutional one.