In short, no, it won't.
Hawking is concerned here about a hypothetical event called "vacuum decay" or a "vacuum metastability event". I would never have the gall to call Hawking incompetent or alarmist, but I do however think that, when provided with room for interpretation (which is in abundance at the frontiers of science and technology) he tends to take a pessimistic viewpoint.
Basically, there is no such thing as true vacuum, even space completely devoid of matter contains a small amount of vacuum potential. This vacuum potential is very important in our day-to-day lives since it informs a great deal of physics at the level of fundamental particles. If it was different, the laws of chemistry, electricity, radiation, gravity, and nuclear forces would be very different.
The hypothesis is that the universe exists at a level of vacuum energy that is not the lowest possible level, but has settled into a "false" vacuum potential state. Like a rock rolling down a hill that has caught on a ledge, it still has some potential energy, but it's being prevented from using all of it up to reach its ground state (the lowest possible potential). However, if one were to give that rock a strong enough push, it would topple over and continue rolling down.
In essence, this is what the vacuum collapse refers to. If our universe has settled into an energy state higher than its true ground potential, then enough energy at any one point could cause it to topple over the proverbial edge and continue rolling down the hill. If enough energy were to be concentrated in the right form in a region of space, it could cause that region to "fall" down to true vacuum. It's possible that there exist stable, tiny regions of true vacuum in space right now that aren't hurting anything due to either not expanding or flashing back out of existence within femtoseconds of forming, and as a future physicist myself I have to say I would really love the chance to research one, if it was somehow possible.
The concern, now, is that if an even greater amount of energy than would be sufficient to revert a tiny region of the universe to true vacuum were generated at a point, and just the right set of circumstances were true, that the region of true vacuum would not just be stable but would begin to nucleate. This would take the form of a bubble of true vacuum expanding into space at the speed of light, inside of which would be a new universe with completely different laws than our own. Needless to say, were such a bubble to consume the Earth, we would not survive the event.
However, such an occurrence is completely hypothetical at this time. Whether or not it's even possible will depend on the precise rest mass of the Higgs Boson, which is not yet known (nor is it even yet clear that the Higgs Boson exists, rather than something mostly like it), as well as the mass of the top quark. This is what's often being misreported as the LHC "Destroying the universe".
And even if it is possible, the amount of energy need to both create would be far, far higher than anything that could conceivably be produced on Earth. And the fact that it hasn't found a way to occur at any point so far in the universe's 13.7 billion year existence should give us a clue to its likelihood of happening now or in the future given that the most energetic events in the universe are, for the most part, behind us.
In fact, the odds of a vacuum decay occurring are so astronomically small as to consider such an event impossible. You are more likely to die due to every single atom in your body undergoing radioactive decay at the same instant in time (yes, it's possible) than due to a vacuum collapse.
There is also an ontological problem that it invokes, namely, that most permutations of this interpretation of the standard model hinge on the many worlds hypothesis being true, in which case we are all subject to subjective immortality and wouldn't need to worry about dying in any way, let alone vacuum collapse, anyway.
So in conclusion, in terms of things that could kill you, worry more about falling down your stairs or getting in a car accident before worrying about the consequences of quantum field theory.
