I have built one strip open canoe and I am now planning on building a kayak. Is it necessary to fiberglass the interior of the boat? It seems to me that the interior fiberglass adds little structural support but significant weight. Would it be possible to just seal the interior of the boat with epoxy and dispense with the layer of glass?

Like Jay said: YES. If you are building a strip kayak the layer of internal glass is absolutely required. The inner and outer skins give the composite section a majority of it's strength. The wood is only a core for the fiberglass.

That said... If you are building a stitch and glue model out of plywood, the full layers of glass can often be omitted as the plywood is stiff and strong enough on it's own (depending on use). Both sides of the seams just get narrow fiberglass tape. The remaining plywood surfaces get sealed with epoxy. It's a very different build method.