Debate as to whether psychology is a science is not uncommon and it doesn't seem to be a debate that will be settled soon. I am aware that there are those both sides of the debate believe the debate is settled.
But instead of the question "Is psychology a science", I think a question we should be asking is, "If psychology is a science, what is it the science of?"
Psychology is often defined to be the science of mental processes and behavior. Those are two very different, but arguably closely related things.
The problem I see is that while behavior is observable in an empirical and objective way (something I see as necessary to consider a field of study a science), mental processes are not in any way observable both objectively and empirically.
In other words, another person's mental processes is effectively as unavailable to empirical and objective study as God is. I believe you think and feel, and have other mental processes, but I have no way of really showing evidence that you do in a way that doesn't require some assumptions.
I may believe God exists, and I may even feel that I talk to God, and I may even believe that God exists to you and that you talk to God, but in a scientific way I can't study God. However, I can study what you say about God, and that would be studying your behavior, and not really studying God.
In a similar way, we don't have access to another's mental processes. They may say that they have mental processes, and I may believe they do through assumption that I don't think anyone would exhibit behaviors without having mental processes. The belief that a person has mental processes requires some assumption, and a belief system that might be called faith.
I'm not doubting that you have mental processes, but I am aware that I can't observe them in any way. And psychology might call itself the study of behaviors and mental processes, but it can only have empirical access to the behaviors of individuals, and I believe that because of that the real scope of psychology is limited to behaviors.
I can easily accept that psychology is a science of behaviors, but I cannot accept that psychology is a science of "mental processes" as long as there is not a way known to observe mental processes and any idea of mental processes in an individual or a group must be inferred.
So if you ask me whether psychology is a science, I might first ask - a science of what?
If you say it's a science of behavior, I would say that's likely as there are many ways to measure behavior. That is what all psychometric exams do.
But it takes a leap of faith to believe that psycometric exams measure beyond behavior and also measure "mental processes". They don't. Any belief that psychometric tests actually measure something like "mental processes" is unproven. We haven't established by theory, any kind of link between the behavior of psychometric tests and corresponding mental processes. To say psychometric tests measure mental processes is unfounded. It's assumed to measure mental processes, it's relied upon, but it is not substantiated by any theory suggesting it is true. It's assumption and faith.