Is the vacuum spacetime around the Earth curved enough to make particles appear spontaneously? I had a discussion with someone who said spontaneous particle creation had been seen in a lab-vacuum. I told him the spacetime around the Earth is not curved strong enough for that. But I have doubts about that. Is it really possible?
 A: 
someone … said spontaneous particle creation had been seen in a lab-vacuum.

This certainly refers to experiments with analog gravity such as sonic black holes. Those are non-gravitational systems that mimic some aspects of curved spacetime physics. So such experiments could detect quanta of analogue Hawking radiation and not a true one, and indeed this has been done, see the Quanta Magazine article for a non-technical discussion of what is the significance of such experiments for gravitational theories.

… the spacetime around the Earth is not curved strong enough for that.

The Earth indeed does not produce Hawking radiation and the reason is not the value of curvature but rather the absence of event horizon. There are some subtleties with dynamical spacetimes that sometimes can emit Hawking radiation without possessing  true event horizon, but for static spacetimes the existence of event horizon is a necessary requirement, and Earth (which could be approximated as a static spacetime) does not have one.
The reason for this requirement comes from conservation of energy. When quanta are produced by static spacetime geometry appearance of a photon outside of an isolated body, carrying positive energy $E$, must be accompanied by appearance of another photon carrying the energy $-E$ so the net energy is conserved. But existence of negative energy photons is possible only for black holes behind the horizon. Informally, the possibility of negative energies for photons inside black holes can be linked with the fact that inside of a black hole time flows along the radial direction, so the energy–momentum component of a future–directed photon along the direction given by the outside time does not have to be positive.
A: The Earth has no event horizon, so it produces no Hawking radiation.
