The number of neurons in the human brain is equated with the number of stars in the Milky Way. Could the Milky Way act as a consciousness of its own, with the gravitational strength between the stars acting as weights of the network?
If we assume the computationalist point of view see e.g. here, then because it's hard to see any nontrivial computations being performed on the local level, you would not think that the galaxy can be a conscious entity in a local sense (i.e. that it generates a consciousness that is experiencing the local environment). But this leaves open the possibility that a consciousness is generated in a non-local sense. E.g. consider ourselves sitting in front of our computers discussing this topic right now here on Stackexchange.
According to the computationalist theory of mind, my subjective conscious experience would be unaffected if my brain were replaced by any other machine that runs the same software. Now, whatever is going on in my brain today is only affected by the things that were going on yesterday but in a sphere with a radius of one light day. So, if you consider that sphere and apply the time evolution operator one second forward, then this is the same as running my brain as it is today one second forward, up to applying the fixed time evolution operator that maps the state of the universe one day forward.
This means that my consciousness right now could be argued to have existed already yesterday, albeit in a non-local form which simply experiences the World as it is now and not the local state of yesterday. By this same logic, you could argue that if the galaxy later would give rise to conscious organisms, then these conscious experiences already exist, albeit that they are subjectively experiencing the future. However, this is then not simply due to the gravity between the stars but also all the relevant physics that needs to be invoked to calculate how these organisms would arise from the initial state.
This looks like it's going to be a "No, it's not possible, because . . ." answer, but hopefully it answers your question. Your proposal is that the Milky Way is a brain, and that the stars are like neurons, correct? I'm not a biologist or neuroscientist, but I do know one characteristic of neurons that helps them do their job: They rely on chemical and electrical impulses.
Neurons work by transmitting electrical signals along pathways called synapses. These signals can be either (nearly) continuous or in regular or irregular "pulses" - i.e. the strength of the signal goes up, down, or disappears. The point here, though, is that signals between neurons are generally not continuous. The gravitational force between two stars, on the other hand, is generally the same, barring a catastrophic event. In other words, the gravitational "signals" from stars don't function like those from neurons.
There are other differences between stars and neurons, too. One is that neurons cannot regrow in significant quantities, whereas stars are forming all the time. Generally in an organism, the neurons in a brain are of similar ages; this is clearly not the case for stars. Another difference is that neural networks are relatively fixed, whereas stars are constantly moving around.
I'll conclude by repeating what I mentioned in a comment, which is that the idea that the universe is a living organism has been around for some time. Another idea is that the universe is actually a giant computer (or just Earth, if you're a sci-fi fan). The problem is that while most people would disagree with these large-scale hypotheses, we may never know for sure if they are right or wrong.
I hope this helps.