Why after replacing the fuel pump, it will not stay running?

Most likely the mechanic followed the directions that state to disconnect the battery when performing electrical repairs. Doing that erases the learned data in the Engine Computer. Most of that related to fuel trims and sensor personalities is relearned as soon as you start driving, without you even noticing. The one exception is "minimum throttle". That requires a very specific set of conditions to initiate the relearn. Until that is done, the engine will not idle fast enough. It may not start unless you hold the accelerator pedal down 1/4". It will tend to stall at stop signs, and you won't get the nice "idle flare-up" to 1500 rpm at start-up.

To meet the conditions for the relearn to take place, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals. At the end of that seven seconds, the computer will take a reading from the throttle position sensor and put that in memory. From then on, every time it sees that same voltage, it will know it has to be in control of idle speed.

That relearn will be cancelled if either pedal is pushed. If the brake light switch is not properly adjusted, a bouncing brake pedal on bumpy roads can cancel the relearn. If you suspect that, hold the brake pedal up with your foot while the coasting procedure is being done.