"Righteous Indignation" is God's wrath directed against sin. Wrath and indignation essentially mean the same thing when exercised by God. Here is a definition of "Wrath" which includes "Indignation" from Nelson's New Illustrated Bible Dictionary:
WRATH — the personal manifestation of God’s holy, moral character in judgment against sin. Wrath is neither an impersonal process nor irrational and fitful like anger. It is in no way vindictive or malicious. It is holy indignation, God’s anger directed against sin.
God’s wrath is an expression of His holy love. If God is not a God of wrath, His love is no more than frail, worthless sentimentality and the concept of mercy is meaningless.
The Bible declares that all people are “by nature children of wrath” (Ephesians 2:3) and that “the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who suppress the truth in unrighteousness” (Romans 1:18). Since Christians have been “justified by His blood, we shall be saved from wrath through Him” (Romans 5:9).
“The day of the Lord’s wrath” (Zephaniah 1:18) is identical with “the great day of the Lord” (Zephaniah 1:14). These terms refer to “the wrath, or indignation, of the Lamb” (Revelation 6:16), Jesus Christ, that will fall on the ungodly at His Second Coming.
How should a disciple of the Lord Jesus Christ exercise "righteous indignation"? We should, like our Heavenly Father, be righteously indignant against sin and all of its ugly consequences.
However, knowing our own weaknesses and proneness to sin, we should be very careful not to direct our indignation against a person but rather against their evil deeds. Hate the sin but love the sinner may be a cliche but it is very apt in this circumstance (see Matthew 7:3-5). Our aim should always be to "convert the sinner from the error of his ways..." (James 5:20), not to sit in judgment. Judgment is God's prerogative not ours and He has delegated that responsibility to His son, the Lord Jesus Christ.
I hope you have found this helpful.
God bless and keep you,