No, this isn't the way a Christian should act. God wants us to be examples of Christ's love and compassion toward others -- but we won't be if we're thoughtless and rude. The Bible says that true love "is not rude, it is not self-seeking, it is not easily angered" (1 Corinthians 13:5).
One of the Bible's most important themes is that God wants to change our lives. First, He wants to change us outwardly -- that is, in our behavior -- by taking away things that are sinful and harmful, and replacing them with purity and goodness. But He also wants to change us inwardly -- in our hearts and minds -- by taking away selfish thoughts and motives, and replacing them with His love and compassion. The Bible tells us to "put on the new self, created to be like God in true righteousness and holiness" (Ephesians 4:24).
We all have "rough edges" to our personalities that God needs to scrape away -- and often it's hard to face them and hand them over to Him. Pray for your cousin, that God will convict him of his insensitive ways, and give him a desire to be more like Jesus. Pray too that, if necessary, God will give you an opportunity to confront him and help him face his need.
But pray also that God will take away the "rough edges" in your life. Perhaps the best way to help your cousin will be to open your own life to Christ's transforming power.