With Galileo, this is sort of a non-question. Galileo (like everyone else of his time) distinguished between the "fixed stars" and things within the local solar system (e.g. planets, moons, comets). But he didn't know much about the fixed stars. By his time it had already been accepted that the "starry orb" (as he put it in a letter) was very far from Earth and much larger than it. In the Ptolemaic view of things, the interior space between heavenly bodies was crystalline in some way. Galileo, in his Dialogue Concerning the Two Chief World Systems, argues that this is unlikely, and rather suggests it makes more sense of the space between was "fluid," but as for what might be in that space, he doesn't clarify or specify what he thinks it is. Satellites (like the moons of Jupiter) already sort of threw the idea of crystalline spheres out the window. They were not the hardest things to throw out, even for those opposed to Galileo.

It should be noted that the idea of a vacuum is probably not what he would have had in mind. The idea that vacuums did not exist was very strongly rooted in Aristotlean philosophy, and to jump from "things move" to "there is literally nothing between them" is a pretty big jump.

It does not mean that he might have thought it was filled with air or gas, though, at least not of the sort you and I think of. There was a notion at the time of "rarified air" or aether, which is to say, an air that is not materially air but is not a vacuum either. If that sounds troublesome, it is, but so is a vacuum without an evidence of one. The anti-vacuumists would say, after Boyle's air-pump was developed, that it might look like there was no air in there, but there is an air that is not affected by material pushing and pulling, or might be much smaller than "normal" air, and this is what they called rarified air. You can kind of thing of it as air's "non-material" part. Again, we don't have any space for non-material air in our current understanding of the world, so this is a strange thing to consider, but recall they also aren't thinking about air in terms of molecules.

An example from the time. Cardinal Bellarmine, a top Jesuit theologian, believed in a fluid cosmos since at least 1572. He argued that there was no crystalline spheres, but it not the incorruptible aether, either. Rather, it "was composed of a fire-like substance that was not incorruptible, could undergo substantial change, and could be easily penetrated." Why come up with something like this? Because comets clearly moved between the fixed stars and Earth/Moon system, so it couldn't be rigid.

To give you an idea, again, of how non-modern Galileo could sound if you actually read what he thought about the stuff we don't think he was right about, here's what he thought was going on with the Sun:

It seems to me that there is in nature a very spirited, tenuous, and fast substance that spreads throughout the universe, penetrates everything without difficulty, and warms up, gives life to, and renders fertile all living creatures. It also seems to me that the senses themselves show us the body of the Sun to be by far the principal receptacle of this spirit, and that from there an immense amount of light spreads throughout the universe and, together with such a calorific and penetrating spirit, gives life and fertility to all vegetable bodies.

Which is to say, there are aspects of that you can warp to make some modern sense of (ok, infrared radiation does matter for things like life), but it's just a very fundamentally different worldview than what we have today. It is not quite so mechanistic, not quite so "naturalistic," even if he is rejecting a lot of the Artistolean and Jesuit doctrines. It has a somewhat mystical feel to it. This is not to criticize Galileo; he did a lot given what he had access to.

So this doesn't quite answer your question but it gives you a flavor for how non-modern the answers at the time likely would have looked. (The sources I consulted on this were Stillman Drake's Brief Introduction to Galileo, which is useful, Galileo's own Dialogue Concerning the Chief Two World Systems, and, for the last discussion of Bellarmine, Mario Biagioli's Galileo's Instruments of Credit, pages 209-210, which are in the context of the doctrine of "fluid skies" and how Galileo's critics felt about it, in particular in response to the question of what sunspots might be.)

By Newton's time, he wasn't really spending time debating Jesuits, so the context had changed a bit. Newton also believed in an aether that permeated the universe and was the rarified medium through which light passed. He referred to the "empty celestial space" between celestial bodies, and was not opposed philosophically to the idea of the vacuum, but I've not seen a lot of discussion that allows me to determine whether he thought it actually was. Apparently in 1706 he considered "what the space that is empty of bodies is filled with" to be one of those big, hard questions that he wasn't sure of the answers. But for all practical purposes, you can say that he thought the space between stars, planets, etc., was empty of material substance (which still leaves an aether or vacuum distinction if you really care about early modern physical philosophy). Newton was prone (following, say, the philosophy of Boyle) to just not worrying about that sort of thing publicly — he tried to concern himself primarily with the measurable and leave what he saw as the indistinguishable philosophical aspects alone, and railed against natural philosophers who took assumptions about atomicity, vacuums, and so on for granted and then tried to figure out things like gravity from those assumptions (e.g., deductive rather than inductive reasoning). This, some critics of this approach charge, was a bit of a bait-and-switch, since to do any kind of inductive work you still had to make big assumptions. When he did hold forth on the nature of emptiness, he was mostly concerned with whether it was empty of God in some way — again, I think, an indication that he doesn't think there's anything material in between the planets. Which, again, I think points to the ways in which they were framing the discussion a bit differently than we would today!

(I find the early modern mind to be about half-recognizable and relatable, and half-alien and foreign. They cared about some of the things we care about today, but there's an awful lot of stuff that is not part of the modern empiricist worldview.)

On Newton, the 1706 quote comes from John Barrow, The Book of Nothing. I also looked at the classic Newton biography, Westfall's Never at Rest, which has the bits about God being in the empty bits (pp. 647). On 17th century debates about vacuums and the kinds of questions scientists should ask, Shapin and Schaffer's Leviathan and the Air-Pump is the classic and essential read.