Absolutely not. Betta's are not known for ripping plants up. During breeding, they might move floating plants around to eliminate hiding places, and to make a spot for the bubble nest among a few strands. They will investigate rooted plants, and the Java Fern where it is attached to a rock or driftwood, but the Java Fern leaves are pretty tough---Goldfish tough. If stem plants, or floating plants rooted in gravel, are in the tank, they will uproot those, but not tear them to shreds.
Another plant choice might be Sagittaria subulatta subulatta (also called Sag. Dwarf Subulatta, or Sag. Dwarf Dwarf). It is a rooted plant, and only grows 2 or 4 inches tall, spreads like a grass in your front lawn, easy to keep under control.
Also another couple of Java Ferns would be good for the small scale of a Betta tank. Narrow Leaf Java Fern, Needle Leaf Java Fern, and Phillipine Java Fern. These are available on Aquabid.com, and Amazon.com
---Mark13