Yes, sun exposure can cause red eyes. Because the UV radiation from sunlight consists of invisible rays (UVA and UVB) that can damage the eye's surface tissues as well as the cornea and lens. What's worse, the UV radiation can burn the front surface of the eye like a sunburn on the skin. If you are exposed, unprotected, to excessive amounts of UV radiation, even over a while, you are probably to experience photokeratitis, an inflammation of the cornea caused by a brief exposure to UV radiation. And it may cause symptoms including red eyes.
A good recommendation is to wear quality sunglasses that offer good protection and wide-brimmed hat when you are exposed in the sun.