No, Obama lacks the authority to repeal “Don’t Ask, Don’t Tell.” This policy was enacted by congress and has since become public law.
Yes, the President is Commander-in-Chief of the armed forces, but this only applies to the use of such forces in times of war:
The President shall be Commander in Chief of the Army and Navy of the United
States, and of the Militia of the several States, when called into the actual
Service of the United States (Article II, Section 2, US Constitution).
Further, it is the power of congress to regulate the discipline of the armed forces:
To provide for organizing, arming, and disciplining the Militia, and for
governing such Part of them as may be employed in the Service of the United
States, reserving to the States respectively, the Appointment of the Officers,
and the Authority of training the Militia according to the discipline
prescribed by Congress (Article I, Section 8).
Likewise, 10 U. S. C. § 654 (“Don’t Ask, Don’t Tell”) states:
(c) Entry Standards and Documents. – (1) The Secretary of Defense
shall ensure that the standards for enlistment and appointment of
members of the armed forces reflect the policies set forth in
subsection (b).
The term “shall” is a mandatory term. The executive branch is given no leeway in the matter.
