This is a great question. Unfortunately, the answer is surprisingly controversial and complex:
The first challenge is figuring out what “objectively true” means. Most –but not all!– people believe there is a universe “out there” that has characteristics which are independent of any particular person’s observations or perceptions. This is a position generally known in philosophy as “Realism.”
To find out if you are a “Realist”, try this thought experiment: Imagine you personally do not exist, and furthermore that other people don’t exist, and in fact, that there is no intelligent life in the universe. Do things such as stars and planets still exist? Does a proton still have the same mass? Does water on earth still freeze at zero degrees Celsius? If you think so, you are a Realist.
The majority of people, even if they aren’t sure that the things that exist actually have the forms and the nature that we think they do, believe that something exists objectively –which is to say, independent of any given subject. Unfortunately, this is impossible to “prove”, given that each of us is imprisoned, so to speak, inside our own experience. There is no way for me to show incontrovertibly that there is anything independent of my own personal experience.
In addition, Realism by itself doesn’t actually tell us as much as you might think –it says something exists beyond our perceptions, but not what that something is. If, however, you believe that the world is more-or-less the way it appears to be, then you are not just a Realist, you are a “Naturalist”, or a “Realist about Perceptions”. In other words, you think the familiar world of trees, sky, earth, stars, moon, sun, etcetera, exists generally the way it seems to.
This seems like a natural (no pun intended) step, but it runs into some problems. One is that it is vulnerable to the possibility explored in the “Matrix” movies that your perception of the world and all its objects are generated by causes quite different than the seeming ones –in the case of the Matrix, a computer, in the case of Descartes’ similar hypothesis, an malicious demon. It may seem far-fetched, but it gains plausibility when we think of how easily our perceptions can be shifted –a set of rose-colored glasses makes everything look pink, just as being told that a student is smart can change our perceptions of his or her classwork.
This leads us at long last to the starting point of your question. In general, scientists replace “Naive Naturalism”, the thought that the world is just as we perceive it, with “Scientific Naturalism”. In scientific naturalism, it is accepted that objects may have a “real existence” very different from our perceptions –for instance, who would guess that fire and water are both made of the exact same building blocks of electrons, protons and neutrons? The claim is made, however, that objective reality relates to our perceptions of it in orderly, rule-governed and consistent ways, and that those relationships can be discovered through the use of the scientific method.
The scientific method is not the only way of discovering truths or “Truths” about the world, but it does have many practical advantages, including the fact that the things it discovers are, by definition, reliable (they will work the same way each time) and fit into a general schema that fits together with all the other scientific truths.
If we take your question at face value, “If something is objectively true, does it have to fall within the realm of science?” the answer is no. Assuming we take a Realist stance, and agree that there ARE things that are objectively true out there, it is clear that some of those true things will be outside the realm of what science can tell us now –whether or not there is intelligent life on other planets, for example –and even that some of it might forever be outside the realm of what science can demonstrate.
I think what you really mean, however, is “does a proposition have to be tested by the scientific method in order to be a valid and reliable piece of knowledge?” That’s not a question that itself has an objective or uncontroversial answer. For example, we might say that there could be other methods –substantially different from our current science –that would give reliable answers. Would we consider those methods “scientific” because they were reliable? In that case, all we mean by “scientific” is “reliable”. But we can’t disprove that other such methods might exist –after all, much of what we consider as good science today would have been alien to scientists of the past.
Another counterclaim is that there are other non-methodical sources of truth. A religious person (such as myself) might be comfortable with the claim that religious revelation provides a source of truths that are equally or exceedingly “objective” as the truths verified by science, while a non-religious person would reject that claim.
As far as the second part of your question, what about things such as art and ethics? Are they intrinsically relativist, which is to say, are they irreducibly dependent on subjective and personal beliefs or characteristics that do not generalize?
On the one hand, we should keep in mind the fact that it might be possible that a “science” of art or ethics could be someday created, which is to say, that we could figure out ways to make statements about those fields that are as testable and reliable as those about fields such as physics and chemistry. In my opinion, however, that would first require not just a different understanding of art and ethics, but also a different understanding of science. Previous attempts to make a science out of such things have generally focused on reducing them to purely material terms, and have not resulted in any notable successes.
This brings up a final issue raised by your question. Do we live in a universe where everything is best explained in purely material terms –i.e. as the extended ramifications of the trajectories of protons, electrons, neutrons and other particles, or even in a universe where everything is capable of being explained in such terms? There are quite a large number of people who would endorse not merely the second statement but also to the first (please notice the differences!). But it should be made clear, such a statement is essentially a declaration of faith. It is not an independently provable proposition, and it is capable of being wrong.
Let me thus rephrase your questions: “Given the current state of human knowledge, is the scientific method the only widely accepted, internally consistent, controllable and reliable way of constructing new knowledge; and do fields such as art and ethics currently fall outside the realm of what science has thus far proven itself to be capable of considering successfully?” The answer to that question is a highly qualified “yes”.