Your friend's claim isn't quite correct for the time slot he gives, but overall the argument that the rocket would "fall out of the sky" because it has a negative vertical acceleration is just plain wrong.
The initial stages of the rocket launch are focused on obtaining a very large vertical speed (at T+125s, some 800m/s), but the rocket's final goal (at least before its eventual trans-Mars injection burn) is a circular orbit at a constant distance from the ground and therefore at zero vertical velocity. That means that vertical deceleration is inevitable and required in the later stages of any rocket launch to orbit; this deceleration is normally enacted by the Earth's gravity sapping the initial vertical speed as the rocket goes up. (Much like a ball that's thrown up on the ascending part of its arc, the fact that it's accelerating downwards doesn't mean it's moving downwards, it just means it's losing vertical speed.)
This process can be seen very clearly on the telemetry data you've linked to: if you go to the velocity vs time link here, you'll see the graph I reproduce below, where the green line indicates the vertical component of the velocity. Note in particular that from T+180s that green line is on a monotonous descent: this is no more and no less than vertical deceleration.
OK, so, having established that the rocket is indeed accelerating downwards, then why doesn't it fall out of the sky? The answer is in that sideways tilt of the rocket: you need a lot of vertical speed to get to orbital height, but what you really need is lots of horizontal speed to actually stay in that orbit.
Ultimately, the answer boils down to the same question for Newton's cannonball, shown below, which does not have a rocket engine and which therefore experiences the full brunt of the gravitational acceleration towards the center of the Earth, but which is going so fast that the ground curves away under it as it tries to "fall out of the sky".
The same thing is happening with the rocket, for which the second stage's cutoff acts like the end of the gun barrel for Newton's cannonball. This is carefully timed with the upwards end of the vertical arc, i.e. when the downwards acceleration of gravity finally gets rid of the last bit of upwards vertical speed, the rocket has achieved so much horizontal velocity that it is no longer useful to think of gravity as a vertical force and you need to think of it as the radial centripetal force that keeps the rocket from flying off into infinity.
For more details, a good resource is the xkcd What if? piece on Orbital Speed.