Congress cannot stop the President from designating another country as a "Major Non-NATO Ally" since the President can designate it through a Presidential Determination. Congress has to be notified as it is mandated under the US Code. Since Congress cannot block the designation, it's likely meant to notify Congress officially. This was the case in 2012, when President Obama designated Afghanistan as a "Major Non-NATO Ally".
As this article by The Hill states:
The Obama administration has 30 days to officially notify Congress on Afghanistan's new status, according to federal law. However, congressional approval is not required before Kabul can assume its role as a major non-NATO ally.
However, Congress can introduce a bill to revoke any country with such a status. There have been bills introduced in Congress that tried to revoke Pakistan’s status as a MNNA, in 2012 and in 2017.
H.R.3000 was introduced in 2017 to terminate the designation of Pakistan as a MNNA and the text of the termination is as follows. It also prevents the President from issuing a new determination to designate that particular country as a MNNA again unless a set criteria is met.
(1) the designation of the Islamic Republic of Pakistan as a major non-NATO ally pursuant to section 517(a)(1) of the Foreign Assistance Act of 1961 (22 U.S.C. 2321k(a)(1)) or any other provision of law is hereby terminated; and
(2) the President may not issue a separate designation of the Islamic Republic of Pakistan as a major non-NATO ally pursuant to section 517(a)(1) of such Act or any other provision of law unless the President submits to Congress a certification described in subsection (b).