One thing to write on a newborn greeting card is to offer the parents congratulations on their child arriving safely in the world and to express hope that the child's young life is filled with happiness and joyful memories. Such a message should end with a message wishing the new family all the best.
Another thing to write on a newborn greeting card is to congratulate the parents in their shift from being only a couple to being the mother and father of a family. If there were no complications with the birth, such a message should also congratulate the parents on the birth of a healthy and happy boy or girl.
Another possible greeting card message is to describe the news of the child's birth as "wonderful" and to express that the parents find parenthood to be both easy and very memorable. This message should end with the writer wishing the family well and volunteering to help with any needs they may have.
If the writer has not spent much time with the mother and father lately, the message may include that he is thinking of them often lately and is looking forward to the next opportunity for all of them to hang out after things have settled down. This message should end by wishing the new parents much love from the writer.