Yes, and it did.
Science is essentially just anything borne out of the scientific method; however, good science typically adheres broadly to scientific objectivity while following that method, which to put it tersely and not completely accurately, says the following:
- True claims are reproducible
- True claims are objective, not based on personal perspective or value judgements
- True claims are backed up by evidence
- True claims are internally consistent
Except for the first, these are not startling concepts. Reproducibility is as such the center of the entire scientific world. If you claim that water is flammable, then you are expected to provide not only your data, but your experimental procedure, so that we can be sure that not only are you being honest, but you didn't screw up the experiment or the calculations.
In fact, reproducibility is how science enforces the other three goals. If others can carry out the same experiment with the same results, those results are objective. The results then are solid evidence. For the experiment to be possible, it must be internally consistent.
So, how does this relate to your question? Because the scientific method was not always in use. Once upon a time, what we now call chemistry was alchemy. What we'd call medicine might have been known as herbalism or other names, depending on the region. Before modern science existed, people still worked within the fields we now identify with solid scientific names like chemistry, physics, and biology. They merely did not follow the scientific method. As soon as science came on the scene, the best parts of those fields we now often look down upon as pseudoscience actually became the foundations of modern scientific fields.
Yet at the same time, while the scientific method might not have been in use, many involved had some idea of what constituted good practice. What sense would it make to claim you had converted lead into gold, if you could not show it? While such extravagant claims appear throughout history, the majority seem to not be taken seriously. The scientific method, while rigorously stated did not exist, is based on a degree of common sense which predated it.
If magic existed, it would be little different. If tomorrow magic suddenly existed in our world, then it would largely be considered absurd and unscientific - yet that is because of centuries of it being so. If it truly was happening, it would not take long for it to be seriously investigated, have the scientific method applied to it, and eventually end up being exploited to menial ends no differently than we exploit quantum physics to play Angry Birds.
Essentially, anything which exists can and eventually will be investigated scientifically. Magic is only in opposition to science in our world because it does not exist - at least by the standards I have laid down. As soon as you can make it reproducible, that changes. That answers the first question in that yes, if you are already performing scientific experiments with it, magic becomes a field of science.
As for your last question, the answer is a little more nuanced. As I said, magic would become another component of science as a whole. It would not replace it, just as quantum mechanics did not replace chemistry, even though it revolutionized physics. However, magic is, as a concept, incredibly broad and potentially powerful, and might have serious implications for other fields of science. If you can destroy entropy, the second law of thermodynamics ceases to hold. If you can teleport, relativity might break down. The exact nature of these implications depend on the exact nature of magic in your world, so I cannot answer them. I can say, though, science as we know it would adapt, not be replaced.