According to Jewish belief, God gives you sicknesses and God gives you cures. If you are sick it's because God wants you to be sick. Once he deems that you no longer need the disease you are then cured.
This can be inferred from the first of the Thirteen Principles of Jewish faith and many other sources:
Belief in the existence of the Creator, who is perfect in every manner of existence and is the Primary Cause of all that exists.
If this is so, why bother with ever going to the doctor?
The same reason that you have to go and earn a living, even though G-d gives you what you need for food, clothing and shelter.
Because G-d wants you to engage in the world and transform it, not live outside it.
To quote the Lubavitcher Rebbe:
We are commanded in our holy Torah, the Torah [of Life, emanating] from “the Living G‑d,” that concerning our health we are to meticulously obey “doctor’s orders,” since “Permission was granted the healer to heal,” and the doctor serves merely as an agent [of G‑d to achieve healing].
Understandably, it is perfectly fine to voice your protests and opinions regarding the doctor’s [prescribed] course of healing — including the notion you wrote to me. However, after the doctor hears you out [and then renders his final opinion], you are to follow his instructions whether you logically agree with them or not.
For, as stated above, the doctor is no more than an agent who heals at the behest and with the permission granted to him by the Torah; [and] since this [power to heal] emanates from the Torah, [the doctor’s orders] are equally beneficial to body and soul.
[The above is true] even when one does not understand the [Torah’s] reasoning [for following doctor’s orders] or thinks differently — which in itself is also not surprising, as Torah is G‑d’s Divine will and wisdom, and thus it is no wonder that not everything the Torah states is comprehensible to man.
However, [it is quite clear that] man must follow all the dictates of the Torah, even when they are not understood — and understanding will eventually follow.
You may be partially correct. Ibn Ezra writes that one is only permitted to go to doctors for external wounds (man-inflicted), but not for internal sickness (God-inflicted) the latter of which are in God's domain for healing.
Exodus 21:19 First Explanation
ולהתיר דברי הרופאים דברי יחיד הם כי לפי דעתי כי האמת להשען ישר דרך על בוראו ולא על בינתו כן בדרך המזלות ובדרך הרפואות כי הכתוב אמר אני י"י רופאך ואין צורך לעשות רופא אחר שותף עמו וכן אמר והסירותי מחלה מקרבך וברך את לחמך ואת מימיך והפך זה אשר לא תוכל להרפא גם כתוב באסא גם בחליו לא דרש את י"י כי אם ברופאים וכן כתוב מחצתי ואני ארפא כי הוא יכאיב ויחבש וטעם ורפא ירפא מהבנין הכבד הדגוש ואיננו כמו הקל והמכה היא מיד האדם ויוכל אדם לרפאותה ומי ירפא שיכה השם רק הכתוב אמר יך ויחבשנו וטעם אשר לא תוכל להרפא כאשר תרפא ממכת בן אדם
Exodus 21:19 Second Explanation
ורפא ירפא לאות שנתן רשות לרופאים לרפא המכות והפצעים שיראו בחוץ רק כל חלי שהוא בפנים בגוף ביד השם לרפאתו וכן כתוב כי הוא יכאיב ויחבש וכתוב באסא גם בחליו לא דרש את י"י כי אם ברופאים והנה הכתוב הפריש כי לא אמר רפא ירפא כי אם ורפא ירפא שהוא מהבנין הכבד ועוד אפרש זה היטב בפרשה הזאת וחכמינו קבלו דברים אחרים עם אלו שניהם כאשר קבלו בשמים עם קטורת הסמים ואינם כתובים
Later he reiterates that one who follows the Torah has no need for doctors:
Exodus 23:25 Second Explanation
והסירותי מחלה מקרבך והנה שומר התורה אין לו צורך לרופא עם השם הנכבד על כן כתוב וגם בחליו לא דרש את י"י כי אם ברופאים
I infer that your question means, "Are you allowed to go the doctor?" rather than, "What's the point of doing it?"
See Shmot 21:19 that states that someone who injures someone else must pay the injured's doctor's bills.
It can be assumed that if the Torah has stated that someone must pay someone else's doctor's bills, it means not only that the Torah allows someone to see his doctor, but, in a sense, REQUIRES that person to see his doctor so that he can be healed. And how do we know that the Torah wants him to be healed? See the beginning of the same verse - "He walks on his crutches".
If you argue - "Oh, G-d healed him and he never saw the doctor", yes, that's entirely possible, but, then, again, the verse obviously does give the injured the option to do so. The point is, that there is nothing implied in the verse stating the person should NOT attempt to be healed by a doctor and leave everything up to G-d.
Maybe God makes us sick because he wants us to go to the doctor? Or pray? It is hard to guess what God wants for us.
What we know is that Rashi (on Bava Kama 85a) answers your question explaining the verse "And heal, he shall heal" to mean we should **not say that we cannot presume to heal someone since God caused him to be sick.
The Rambam (in Deot 4:23) writes clearly that a talmid chacham should not live in a city without a doctor.
Arie Pelta here also writes
In Sefer Devarim, the Zohar (Haazinu 299a) states, “one should not say since G-d imprisoned, one should not try and release him”. Even though G-d had imprisoned the sick man, nevertheless, one should try to free him. If the doctor can cure his physical illness, good. If not, the doctor should try to bring him to repent from his sins and cure his soul.
To take the counterpoint, the Ramban writes (Vayikra 26:11 towards the end), as in your question, that sick people should turn to God and not doctors to heal them. How can it be knowing the Ramban was also a doctor? There are a number of explanations, e.g., the Ramban speaks of a utopian society (artscroll on Brachot 60a), or he speaks of "hidden diseases" (e.g., psychological) which in his time were not healable as of today (Ibn Ezra, see also here for more on this Ramban).
In any case, the halacha (Birkei Yosef on YD 336:2) says clearly one cannot rely on miracles and should go to doctors. See also here regarding another Ramban forcing someone with a life-threatening condition to ask others to desecrate Shabbat to save his life.
