Facilitate means to make easy or easier: “Winning millions in the lottery may facilitate entering the world of high-class society”
Felicitate, on the other hand, carries the following meanings:
1. To wish joy or happiness to: “The parents of the bride wanted to felicitate the newly wedded couple before they left for their honeymoon.”
2. To congratulate: “The principal decided to felicitate the student who got the highest marks.”
I think if I call my friends instead of writing to them, it will facilitate my ability to felicitate them for their new baby.