What natural plant/herb is Viagra chemically extracted from?

It wasn’t necessarily extracted directly from a plant source. Scientists at pharmaceutical companies systematically study molecules by the millions. Some of which were taken from plant sources and some where not. Most of them are modifications from previously studied molecules. They might extract a substance from a plant, analyze its molecular composition, then find one molecule that shows promise and warrants further study. Then they might find that the molecule isn’t desirable for some reason, so they might slightly alter the molecule, then test the new molecule that was formed, and so on. They literally study and discard millions of molecules before finding the one that demonstrates a useful medical application. With computers this process is much faster than it used to be.
In the case of Viagra, sildenafil citrate was originally being developed to treat ischemic heart disease. During human studies, they found that sildenfil didn’t demonstrate the promise they had hoped for, but in gathering feedback from the patients on side effects, they stumbled across the “side effect” of a hard erection in patients that hadn’t had a “good erection” in years. They decided to study it further and found it was very effective for this purpose. They realized that erections are regulated by an enzyme in the body called phosphodiesterase. Variations of this enzyme are found throughout the body effecting various things, and are numbered for classification purposes. The phosphodiesterase in the penis is called “PDE5″. Other enzymes like PDE11 are found in skeletal muscles. In a nutshell, PDE5 prevents an erection by gobbling up nitric oxide. Nitric oxide is released during arousal to cause the erection, but impotent men have too high a threshold of PDE5 to allow it to happen. Viagra, Cialis, and Levitra all work by temporarily inhibiting PDE5, thus nitric oxide is released during arousal and is allowed to cause the erection. The differences between the drugs is due to the levels at which other PDE’s are inhibited. Viagra also inhibits PDE3 which is found in eyes, relating to color vision, which is why you can sometimes have a temporary bluish tint to your vision. And Cialis inhibits PDE11, which causes an increased incidence of sore muscles. The drugs also differ with regard to their onset of action, and half life.