No, California is not going to fall into the ocean. Despite the geology and physics that only exist in Dwayne Johnson's films, this won't happen for several reasons. The average annual rate of Pacific Plate films relative to North American plate is 46 mm per year to the northwest (similar to nail growth rate). This is an average plate movement, while a large earthquake on the San Andres fault could trigger significantly greater localized movement of the order of tens of feet in seconds.
The moving nature of the San Andreas fault system means that Los Angeles will one day be adjacent to San Francisco. However, it doesn't allow a massive fall or westward movement to be required for California to fall into the ocean. This movement will continue to drag parts of California to the north. However, the plate boundary follows the western coast of North America and, therefore, any land mass trapped on the western edge of the plate boundary will be dragged north along the west coast of North America.
But while the Great One would definitely cause mass destruction, it wouldn't sink part of California into the ocean or separate the state from the rest of the country. The idea arises from a misunderstanding of the seismic forces that cause earthquakes in the region. Powerful earthquakes occur frequently along the west coast of the United States because the region is close to a boundary between two tectonic plates. If you've read How Earthquakes Work, then you know that the Earth's surface is made up of large, rigid plates that move slowly over the mantle layer below.
On the boundaries between the plates, several things can happen. The Pacific plate and the North American plate simply move against each other: one slowly creeps to the northwest and the other to the southeast. Friction builds up along faults because the two sides come together very tightly. If the frictional force exceeds the forces that move the earth, the two sides will lock, so they will stop crawling.
When this occurs, stress builds up along the fault line until the force of the movement is large enough to overcome the frictional force. Then, the pieces of land suddenly snap into place, releasing a large amount of earthquake-causing energy in the Earth's crust. Many scientists estimate that there is enough tension built up along some blocked California faults, that when they finally slide, the earthquake will be extremely powerful. The Hayward Fault particularly affects these scientists because it is located under densely populated areas in and around Los Angeles.
The notion that part of California will separate was likely inspired by the San Andreas fault. After all, since the fault runs through California, one part of the state is on the Pacific plate and another on the North American plate. If those plates move in different directions, it makes sense that the two parts of California would also move in different directions. And this is, in fact, the case.
But, even in a massive shift along the fault, the plates travel an incredibly short distance, a matter of feet at most extreme changes. Tension cannot build up to a point where an entire mass of land moves many miles relative to another, so you won't see any substantial piece of land separating from another. Instead, pieces of land will move away from each other very slowly over millions of years. One end of California can drift slowly until it is finally underwater but this can hardly be interpreted as sinking in an ocean.
Despite rumors that a major earthquake will cause California to break into an ocean, there is nothing to worry about as this is impossible to occur from an event on San Andres fault due to its type of fault. The San Andres Fault System which crosses California from Salton Sea in south to Cape Mendocino in north is boundary between Pacific Plate (which includes Pacific Ocean) and North American Plate (which includes North America). The movement of fault that impacts California and slides it northward is direct result of two massive plates moving in different directions relative to each other. Most Southern California faults are course slip meaning they move approximately parallel to each other in opposite direction with little vertical movement.