In Romans 7:25-8:2, Paul said that he served the Law of God with his mind in contrast with saying that he served the law of sin with his flesh and he said that the Law of the Spirit has freed us from the law of sin and death, so the law of sin and death is not the Law of God whereas the Law of Moses is the Law of God (Deuteronomy 5:31-33). All of God's righteous laws are eternal (Psalm 119:160).No. Being a doer of the law, because it is the law, doesn't, and can't, save, but will only bring about God's wrath. Christ saves us solely by saving us and with nothing beyond that either required or permitted, which makes Christ alone the Savior, period. The law (as you understand the law), isn't the law that is in view for salvation. The law in view for salvation is the law of the Spirit of life in Christ, not the law of Moses, which law was done away with in Christ as we have been so informed in Hebrews.
[2Co 1:19-22 KJV]
19 For the Son of God, Jesus Christ, who was preached among you by us, [even] by me and Silvanus and Timotheus, was not yea and nay, but in him was yea.
20 For all the promises of God in him [are] yea, and in him Amen, unto the glory of God by us.
21 Now he which stablisheth us with you in Christ, and hath anointed us, [is] God;
22 Who hath also sealed us, and given the earnest of the Spirit in our hearts.
[Rom 8:2 KJV] 2 For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death.
Our salvation from sin would be incomplete if we someone were only saved from the penalty of their sin while they continued to be directed at being doers of sin, so there is an aspect of our gift of salvation that we are experiencing in the present by repenting and redirecting our lives towards being doers of the Law of God. We can't earn our salvation even as the result of having perfect obedience to the Law of God because it was never given as a way of earning our salvation in the first place, but rather it was given in order to teach us how to experience God's gift of salvation. The content of a gift can be the experience of doing something, such as giving someone the opportunity to experience driving a Ferrari, where the gift intrinsically requires them to do the work of driving it in order to have the experience of driving it, but where doing that work contributes nothing towards earning our salvation. In Titus 2:11-13, the content of our gift of salvation is described as being trained by grace to do what is godly, righteous, and good, and to renounce doing what is ungodly, so doing those works in obedience to the Law of God has nothing to do with trying to become saved as the result, but rather we are intrinsically required to be a doer of those works because God graciously teaching us to experience being a doer of them is part of the content of His gift of salvation.