The Christian way is encapsulated in statements such as "Judge not lest you be judged" and "Let he who is without sin cast the first stone". In other words, the Christian way is to leave these matters to the law (if and where applicable) and to the Christian God in heaven.
The first quotation is from a series of moral guidelines taught by Jesus (Matthew chapter 7). In the second of these quotations from Jesus (John 8:7), the Pharisees sought to tempt Jesus to rebuke the young woman for her adultery. Confronted by Jesus' rebuke, they skulked away and left her alone.
The verses below would have to be some of the most mis-quoted and mis-understood verses in all of scripture. They do not at any time say not to judge, but they do insist on a person's attitude being right before they do so. In which case, if the attitude is not right, then don't do it.
These verses also highlight the need for humility and a lack of hypocrisy in dealing with sin in the lives of others.
The use of the term 'brother' also implies strongly that the person who is to do the rebuking has the 'right to speak'. That is, they are not a stranger but they know and care about the person they intend to rebuke, and that this is recognized by both parties. Otherwise better to stay silent.
Any rebuke that would be given, in order for it to be done 'the Christian way' would also contain genuine love for the person and concern for their eternal well-being. This is not done in a patronizing or proud manner either (which would not be so anyway if the verses above are truly taken to heart). Being non-judgmental and loving does not alter the fact of sin, whether it be pride or homosexual behavior. Another thing that doesn't go astray is listening. Many people are hurt and need healing - Jesus can provide that when the need is felt and this can never be imposed. On the other hand, denying the seriousness of sin, any sin, does not help but only hinders matters.