"Seeking to establish his own righteousness" is trying to be righteous by your own works and law-keeping which you are promoting as the Jews have done since Moses.
"Submitting to the righteousness of God" is submittng to
the righteousness of God without the law [as] manifested, being witnessed by the law and the prophets; even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believeRom 3:21-22.
So that NOW under the New Covenant of the Gospel of the Grace of Christ,
Everyone who calls on the name of the Lord will be savedRom 10:13.
CHRIST is the ONLY issue.
This is typical antinomianism that you are spouting, and part of the reason why the modern church looks so much like the world.
Many will think that they "have called upon the name of the Lord". But the truth is they have not.
21 Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven. 22 Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works? 23 And then will I profess unto them, I never knew you: depart from me, ye that work iniquity. (Matthew 7:21-23)
Jesus says they must "do the will of the Father."
They didn't lose their salvation. They never had it. Jesus says, "I never knew you", not "I used to know you."
28 And even as they did not like to retain God in their knowledge, God gave them over to a reprobate mind, to do those things which are not convenient; 29 Being filled with all unrighteousness, fornication, wickedness, covetousness, maliciousness; full of envy, murder, debate, deceit, malignity; whisperers, 30 Backbiters, haters of God, despiteful, proud, boasters, inventors of evil things, disobedient to parents, 31 Without understanding, covenantbreakers, without natural affection, implacable, unmerciful: 32 Who knowing the judgment of God, that they which commit such things are worthy of death, not only do the same, but have pleasure in them that do them.
The so-called professing sodomite who continues in his filthy behavior has been given over by God. There may be some who have engaged in sodomite behaviors and then stepped away. But according to Romans 1, it is rare sodomite who can come to God. True, some few have (see 1 Corinthians 6:11), but the overwhelming majority are Romans 1 reprobates.