Paul says in Romans that we should 'Be careful to do what is right in the eyes of everyone'. (Romans 12:17. NIV). However, I would certainly not characterize Paul as being 'politically correct' by any stretch of the imagination. It made me wonder they sound like the same thing, so how is doing 'what is right in the eyes of everyone' so different from being 'politically correct'?
I think this is a translation issue (so possibly better at BH SE).
Look at the Amplified Translation of Romans 12:17:
Repay no one evil for evil, but take thought for what is honest and proper and noble [aiming to be above reproach] in the sight of everyone.
The NIV calls this section (v9-21), "Love in action" which I think summarises Paul's thoughts here well. Therefore this is about acting properly toward everyone - not taking on their morality but rather upholding God's standards before everyone.
This whole chapter seems to be talking about the opposite of political correctness. In verse 2:
Do not conform yourselves to this age but be transformed by the renewal of your mind, that you may discern what is the will of God, what is good and pleasing and perfect.
In verse 9:
Let love be sincere; hate what is evil, hold on to what is good;
In verse 14:
Bless those who persecute (you), bless and do not curse them.
Have the same regard for one another; do not be haughty but associate with the lowly; do not be wise in your own estimation. Do not repay anyone evil for evil; be concerned for what is noble in the sight of all. If possible, on your part, live at peace with all. Beloved, do not look for revenge but leave room for the wrath; for it is written, "Vengeance is mine, I will repay, says the Lord."
All of this, and probably more, taken together, talks about being sincere in what is good and evil, but treating others with compassion, love, and dignity when regarding them. The "if possible" verse, has to do with how you treat others. If you regard someone who has done evil in a haughty, better than you, disrespectful way, then you are guilty of not doing all that is possible to live at peace with that person.