Police were called to the bridge shortly after 12.35pm, where they found two men in the water who were both pulled out with the help of an RNLI boat.
A man, aged in his 40s, was pronounced dead at the scene by ambulance staff.
The second, also aged in his 40s, was taken to a west London hospital, where his condition is stable.
He has been sectioned under the mental health act.
A witness, who did not want to be named, said he saw the pair sitting on the edge of Kew Bridge before they entered the water.
Another man, who also did not want to be named, said he saw them swimming.
One “tried to get to the side”, he said, while the other swam downstream.
A London Ambulance Service spokesman said: “We were called at 12.37pm today to reports of an incident on Corney Reach Way, Chiswick.
“We sent two ambulance crews, an incident response officer, two medics in cars and an advanced paramedic to the scene. We also dispatched our hazardous area response team and London’s Air Ambulance.
“We treated two men at the scene. Sadly, despite the efforts of medics, a man died at the scene.
“We took the second man to a major trauma centre as a priority.”