Does walking by the Spirit in the New Covenant negate the Law in sanctification? Are the Spirit and the Law antithetical?

Short answer, no.

Long answer (What Law?),

There exists, apart from Sinai (written code), the “law of the Spirit of Life” (Romans 8:2; Romans 2:14-15; Romans 7:6), it is the Spiritual Law written upon our heart at conception that has been extant as an arbiter of free will subsequent the creation of the angels…

…that same Law judged Satan of covetousness (Isaiah 14; Ezekiel 28; John 16:11, Revelation 20:10)…

…and that same Spiritual Law was infused within the human genome on Day-6 (Genesis 1:26-28) and judged the Adamic/Noahic generations of defilement worthy of extinction save eight (Genesis 6).

That same Spiritual Law provided a foundation for the Code of Hammurabi and the Sinai Commandments (written code) given the children of Israel; this, due to concerns relevant to Israel’s compromise with paganism serving pagan gods.

That same Spiritual Law is used by the Holy Spirit in the New Covenant to sanctify the redeemed in Jesus as the Spirit produces fruits of righteousness through the redeemed conscience that glorify the Father and His Kingdom and validated, authenticate, saving faith (Galatians 5; Matthew 7:15-20)…

…and that same Spiritual Law will be used by Jesus to judge the disobedient and unbelieving in the Judgment of the Condemned (Revelation 20:11-15)…

…and that same Spiritual Law will be extant in the New Jerusalem as free will remains extant in Eternity by necessity.

You are correct that the Christian is not under Sinai’s law concerning righteousness (a right standing) with the Father as per New Covenant promise (Romans 6:14) but the Spiritual Law will forever play a role in our relationship with our fellowman here on Earth, in the resurrection and with our Creator throughout Eternity.

It is true that righteousness (a right standing) with the Father does not manifest through vain attempts to keep the Mosaic Law, Sinai, a written code (Galatians 2:16; Romans 3:20) but the law of the Spirit of Life will eternally be relevant as an arbiter of free will by necessity as free will is an essential attribute concerning intimacy and genuine, reciprocal, relationship with our Creator.