Is there anything stopping you from just staying on an island?

If you stay on an island, you won't advance the story. Advancing the story requires exploring the world.

You'll also eventually run out of food and crafting resources. Resources such as wood, fruit, and scrap are finite on an island. Once you've harvested everything from the immediate area (island, sea, sea floor), you must set sail to gather more resources.

Toward the later stages of the game, you're able to farm some resources (such as wood and fruit). You and also catch, cook, and eat fish. However:

• Fishing poles require wood & leaves, and chopping down trees requires stone or scrap. You'll eventually run out of stone and scrap if you aren't pulling in barrels or visiting new islands.
• Fruit yields fewer seeds than you used to plan them. Watermelons and pineapples only yield a seed 2/3 of the time when harvested. Mango trees yield a seed 14.3% of the time with every chop, and palm trees yield a seed 11.8% of the time with every chop. At these rates, you'll eventually run out of seeds unless you continue visiting new islands.

References:

• Mango Seed (via Raft Wiki)
• Palm Seed (via Raft Wiki)
• Pineapple Seed (via Raft Wiki)
• Watermelon Seed (via Raft Wiki)
• Tree farms are totally worth it, but I've been struggling with sustainability of seeds. Tips? (via Reddit)
• Sustainable gardening? (via Reddit)