I imagine you are thinking since you are no longer together, shrugging off Father's Day seems like a no brainer. But an end in a relationship does not signal the end of your co-parenting duties with your ex.
Like it or not, co-parenting means you should acknowledge Father's Day with your children. Even though you and your ex are divorced or no longer together, since you have a child in common your ex will always be in your life. You will see your ex at birthday parties, sports, graduation, your child's wedding...just to name a few occasions.
And if you are in a child custody dispute, one of the things the court looks at is if the parent encourages his or her child to have a loving, stable, and nurturing relationship with the other parent. Maintaining the parent-child bond in a child custody dispute, is what the court considers to be in the child's best interest. And what is in the child's best interest is the court's overriding concern when entering in a parenting plan. Parents who encourage and maintain their children's relationship with the other parent are acting in their children's best interest and may receive a more satisfactory parenting plan.
You do not have to call your ex to wish him a "Happy Father's Day," but you should have your children write him a card, acknowledge to them that it is daddy's special day, or encourage them to do something to show their father their love. The bottom line is your chlidren have two parents whom they love with all their hearts.
Here is a article for great insights on this topic. 9 Reasons to Appreciate Your Ex This Father's Day.