If you could make it down to the core, then yes, you would probably be able to experience a "solid surface" (where I put that in quotes for reasons that should be apparent in a moment).
The question really gets to, though, what you consider to be a "solid" and a "surface" in a gas giant. The issue at hand is what the Wikipedia article stated - as you go down through the atmosphere, you encounter denser and denser material. Gas under incredibly high pressure will start to behave like a liquid, and be as dense or denser than a liquid such as water. There would be no definite point at which you could say the stuff above you is clearly "air-like" while the stuff below you is clearly "water-like," it's a gradient.
You would also get crushed long before you made it anywhere near the core, just like it's only fairly recently that we've been able to build submersible vessels that can go to the deepest parts of the ocean on Earth.
This might not fully answer your question, but maybe it will be a good start.
Things to consider
- Thermal energy received by Jupiter from the sun
- Thermal energy radiated by Jupiter (hence, net thermal energy)
- Jupiter's composition
- Jupiter's temperature
- Jupiter's gravitation
- Hydrogen's thermal properties (among other properties)
For the first 2 items to consider, we actually don't have much of a problem. We could go about figuring out the sun's radiation power (well...on wiki it says $3.846 \times 10^{26}$ if you're interested), estimate the energy received by Jupiter by making a surface area argument and then figuring out Jupiter's radiation power. From this site http://www.tritonfun.com/custom.em?pid=594668, it actually says that Jupiter is radiating 1.9 times the amount of heat it receives from the sun. But this is because it is also creating it's own energy from a variety of methods (including radionuclides). We can ignore all of that though because I'll bet that Jupiter is close to thermal equilibrium in the short term (ie, it's not constantly changing temperature as a whole extremely drastically). If you weren't given the information as to how hot a planet was, this is the kind of analysis you would have to perform in order to get the average temperature of the body. Luckily we can use data from other sources for the temperature of Jupiter.
That addresses points 1 and 2. From a number of online resources, Jupiter has multiple layers, namely, the outer, gaseous atmosphere, a transition area between gas and liquid, a liquid/metallic section and the a mostly hydrogen core. Unfortunately I can't find exact distances of all of these...which would make answering this much easier. But, what we need to know is that the layers differ drastically in terms of pressure and temperature. In fact many layers that are farther from the center of Jupiter are actually hotter, but with less pressure. So the pressure and the temperature combined will affect what phase hydrogen is in. It's extremely complicated and probably well beyond the scope of most physicists to understand why some layers have more pressure than others, but we could just make a vast simplification and say the pressure and temperature increase/decrease as a simple function of distance from the center of Jupiter.
Jupiter's temperature and pressure increase steadily as we get closer and closer to the center only at the core. Near the phase transition region between gas and liquid, the temperature is about 10,000 K and the pressure is 200 Gpa. The temperature at the core's edge is around 36,000 K and 3000-4500 Gpa. So it seems like both temperature and pressure increase throughout the liquid layer/transition layer too. I'm not necessarily sure how to calculate this from scratch, given the elements, the mass of the planet, it's radius and density. Of course it will involve balancing the forces of gravity with electrostatic forces (I just tried a quick calculation using those 2 forces only, but it was WAY off). The fact that my calculation was pretty far off tells me there are a lot more things you have to consider that are extremely important for accuracy. For example, as the hydrogen molecules heat up, they will be bouncing around faster and faster, causing a greater repulsion. Details like this make the calculation very difficult.
I know this doesn't help you a whole lot, but hopefully I've illustrated some important points. Firstly, this kind of calculation (if you want it to be accurate) is something people with PhD's who specialize is this kind of stuff calculate. You can make huge simplifications and maybe get an okay answer. But I feel like it's a wasted effort to actually go through the calculations if they don't give you a very accurate answer. But like I said, if you want to get a rough answer, think about all of the forces going on. Gravity, electrostatic repulsion, average thermal energy, and then make the net force = 0. Electrostatic repulsion of hydrogen can be found on the wiki page and hopefully you know how to take into account gravity as a function of distance from the center =p.
Best Answer
I'm seeing a lot of references to Roche limits and the like in the answers here, but to me, that is implying a far slower process than what I would think would be described as a "collision". Just assuming a collision velocity no greater than that starting the two planets at rest and letting them fall together under their own mutual gravity, they're going to be going so quickly by the time they collide that there certainly won't be time for them to break apart due to being within the Roche limit.
If we take this question as Earth colliding with Jupiter, then technically, theoretically, yes, Earth would begin to break apart, but we're talking about a minimum collision speed here of about 60 km/s (Jupiter's escape velocity). The entire collision at that speed, from first contact of atmospheres to centers-of-mass coinciding, is only about 20 minutes. So while technically one or both would begin to break apart, the differential acceleration between their nearest and furthest points will be on order of about 4 m/s², making the total breakup on order of about 4800 km, or around a third of the diameter of the Earth. Not insignificant, but fairly trivial in terms of the overall impact's effects. (Consider the difference between a cannonball splashing into a barrel of water vs. a dumbbell. Yeah, technically they'll behave somewhat differently, but in terms of the level of catastrophe, it's pretty trivial.)
Far more important is to consider that at the magnitudes of forces and velocities involved in a collision like this, everything behaves as a fluid. So yes, the analogy of it being closer to imagining two spheres of water colliding than two solid objects is quite apt.
Interestingly, there's a theory that almost exactly this happened in Saturn's past: a roughly Earth-sized object may have collided with it in the distant past. This did not, of course, result in Saturn breaking up, but it may have resulted in 'splashing' its core into something quite a bit less orderly and more 'fuzzy' than suspected before. It may even still be 'sloshing' in a sense, creating some of the resonances responsible for the specific patterns of its rings!
I'm also seeing a lot of assumptions of velocities quite a bit lower than Jupiter's escape velocity, but... how would this happen? If we put ourselves in a Jupiter-centric frame of reference (but non-rotating), then the least-fast object that could conceivably collide with it would be one that is stationary at infinity, i.e. at its escape velocity. In order for a colliding object to be going slower than that, it would need to pop into existence at some finite distance from it, which can't really happen outside a simulation.
So, combine the minimum speed possible for this scenario (60+ km/s²), the short duration of the collision (1200 seconds or so), and the sheer forces and amount of heat involved, and what you'd have is more of a 'splash' of a smaller sort of egg-shaped ellipsoid of water hitting a much larger and more massive sphere of water.
For an intuitive sense of what it might look like, check out some simulations of the collision that is theorized to have created the Moon, such as: https://youtu.be/o2lRpiediP8?t=340 This would be quite a bit less dramatic, with 300:1 mass difference instead of 10:1, but still enough to hopefully give some idea.