The short answer is: no.

To understand why this is it is helpful to remember two things.
The first is that light is a wave. This wave is partially electric and partially magnetic, with each color corresponding to a certain wavelength.
The second is that your eye sees because the electric fields that make up the light wave cause the electrons in your eye to vibrate. However, only some wavelengths of light can make the electrons in your eye vibrate enough to see. These wavelengths are the wavelengths in the visible spectrum (red, green, blue, etc.) Light with longer wavelengths (radio waves, micro waves, infrared, etc.) can't make the electrons in your eye respond, and neither can light with shorter wavelengths (ultraviolet, x-rays, gamma rays, etc.), so you don't see them. So the slightly longer answer to your question is that your eye is really only sensitive to electric waves (i.e., light), and even then only to electric waves within a specific range of wavelengths (i.e., the visible spectrum).

Gravity waves aren't electric waves and so aren't very good at making electrons vibrate (gravity is very weak in this sense). In addition, it is very hard to make gravity waves with the wavelengths that your eye would notice. However it is true that if you were to make incredibly strong gravity waves at the right wavelengths the electrons in your eye would respond and you would see the gravity waves, just as you do light waves. But to do this would be very nearly impossible.

By the way, the reason that you "see" lightning is because the flow of electricity through the air (the lightning bolt) makes the air so hot that it glows - i.e., it gives off light. So it is not really the electricity you are seeing, but rather the hot air created by the flow of electricity.

I'm sure you know from you're basic science classes that electricity is carried by moving electrons. In the gases that make up air, these electrons are normally strongly attached to the molecules that they form. However, during a lighting strike, they're ripped away and can move about, allowing electricity to flow. This leaves behind positively charged molecules. The mix of electrons and positive molecules is called a plasma. When the negative electrons recombine with the positively charged gas to re-form stable molecules, visible light is given off. That's what you see.

So, the visible light that you see from a lightning strike is the same as the visible light that you see all around you each day. You are not actually "seeing" electricity, rather you are seeing evidence of electricity.

Likewise, you can't "see" gravity. However, if you go to the ocean's edge, you can see the tide rise and fall. This is a result of the moon's gravity pulling on the earth. So, yes, just like the visible evidence of electricity you see in a lightning strike, visible evidence of gravity is all around you. Hope this answers your question.

When you see light of any kind, be it from a fluorescent lamp or from lighting in a thunderstorm, you are seeing the same thing. We cannot separate "electricity" from "light", because light is actually the joint vibration of an electric and a magnetic field. We call that vibration an electromagnetic wave. The retina of our eyes is sensitive to electromagnetic radiation within certain energy range. The gravitational interaction, as far as we know today, is not of electromagnetic nature, therefore there is no visible form of gravity.

That is actually an extremely complicated question you ask. But basically the answer is: We are not really sure.

Physicists are constantly doing research on the subject to try to better understand gravity. From what we understand now, the force of gravity is so fundamentally different from the electromagnetic force (which light is a very small part of) , that there is no way we can directly "see" gravity itself. We can however, see the effects of gravity in things like black holes. A black hole is an object with so much gravity that nothing can escape from it, not even light. So in this way we can see the effect of gravity acting on light. Hope this helps. I would be glad to clear up anything that may still be confusing.