There is no such thing as a gadol.
Pirke Avos exhorts us to "aseh l'cha rav", make for yourself a teacher.
Each and every Jew must choose his/her own trusted, knowledgeable, halachic poskim.
A great example of this is the "shabbos mode" in modern electric ovens. (a misnomer, since it is really for yom tov use.)
Rav Shmuel Heinemann worked with electric oven manufacturers to create a system by which one could raise and lower the oven temperature on yom tov.
A few years ago, right before shavuos, someone approached Rav Eliashiv (held by some to be a "gadol") and asked him if these ovens could be used on yom tov. It's never clear exactly how the question is asked to the "gadol", what information was and was not conveyed, etc. In any case, Rav Eliashiv ruled that it is forbidden to use these electric ovens on yom tov.
Rav Heinemann held to his original psak din.
We don't have an authoritative central Jewish court today. There is no hierarchy of poskim. Learn Torah, don't shop for the answer that you want, but pick poskim that you are comfortable with. Rest assured that by following their rulings, you are living a halachic life.
If you meant to ask, "Can I argue with my posek if I know that I'm right" - it depends on the relationship that you have with any given posek.
If you call up a she'elah hotline, the posek on-call is not going to give a shiur on your subject. He will likely say "mutar" or "assur", and then move on to the next caller. If you have a personal relationship with your posek, then you can bring in additional sources from shas and ask how they would apply to your case.
Finally, one can also understand aseh l'cha rav to mean, "Make unto yourself a teacher". If you have a thorough knowledge of shas + poskim on the specific topic of your question, it is legitimate to be your own posek.
However, many leading rabbanim and poskim try to avoid ruling on their own life questions, since they are de facto nogea b'davar, and may be misled in their conclusions. Rather, they will defer to a posek of their own choosing, other than themselves.