Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Brother, thank you for your good question and interest to know more about Islam and its teachings.
Generally speaking, a Muslim is required to show kindness, love, and dutifulness to his parents. Father’s Day is a social and family event that aims at celebrating fatherhood and honoring fathers. Such social and cultural events are not rejected in Islam as long as they don’t promote un-Islamic values.
Responding to your question, Dr. Wael Shihab, PhD in Islamic Studies from Al-Azhar University, and currently the Imam of the Downtown Toronto Masjid in Canada, stated,
Dutifulness to parents, both mothers and fathers, is an Islamic value and obedience to Allah. One, therefore, should show kindness towards his parents because Allah Almighty says, “Your Lord has decreed, that you worship none save Him, and (that you show) kindness to parents. If one of them or both of them attain old age with thee, say not ‘Fie’ unto them nor repulse them, but speak unto them a gracious word.” (Al-Israa’ 17:23)
Ibn `Abbas (may Allah be pleased with him) reported that the Prophet (peace and blessings be upon him) was once asked about the best deeds: “The Messenger of Allah (peace and blessings be upon him) says, ‘Having faith in Allah and His Messenger, then honoring one’s parents.” (Al-Bukhari and Muslim)
In fact, the core meaning of the prohibited bid`ah in Islam is to introduce or invent a devotional or ritual act of worship into religion with no backing from the Shari`ah. It is not acceptable, therefore, for anybody to invent a new form of prayer or fasting. `Ibadat (acts of worship and religious rituals) should be founded on correct Shari`ah proofs.
World Days and events—such as the World Hijab Day, Mother’s Day, Father’s Day, and Orphans’ Day—are mostly based on cultural and social considerations. They are not meant to introduce new religious rituals or devotional acts of worship.
Father’s Day is a celebration honoring fathers and celebrating fatherhood, paternal bonds, and the influence of fathers in society. Many countries celebrate it on the third Sunday of June, though it is also celebrated widely on other days by many other countries. Father’s Day was created to complement Mother’s Day, a celebration that honors mothers and motherhood.
Given the above, Father’s Day is a social event that aims at supporting universal values and morals. It’s not objectionable in itself as long as honoring fathers—or parents—is not confined to that Day. It’s not a sinful bid`ah as it’s not meant to introduce a new ritual or `ibadah into religion. It’s a cultural, social, and family Day that aims at maintaining social and family ties.
May Allah help you and accept your and our good deeds!
Allah Almighty knows best.