No, it's not magical
The Sage Advice Compendium, in the question "Is the breath weapon of a dragon magical?", lays out a series of tests for whether or not a particular ability is magical for the purposes of the game's rules:
- Is it a magic item?
- Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
- Is it a spell attack?
- Is it fueled by the use of spell slots?
- Does its description say it’s magical?
In the case of Blessed Strikes, the answer to all of these questions is no; it's not a magic item, is not a spell or replicating a spell's effects, doesn't involve a spell attack, doesn't require spell slots, and the words "magic" or "magical" don't appear in the ability's description. Therefore, Blessed Strikes are not a magical effect for game rule purposes and still work fine in an antimagic field.
As Sage Advice explains, the rules draw a distinction between what it calls the "background magic" of the D&D world and the "concentrated magical energy" that is involved in magic items, spells, and similar magical effects. As far as the rules are concerned, your cleric's ability to dish out radiant damage is just a part of how the world works and is no more magical than the ability of dragons to fly and breathe fire or whatever else.