It's grammatically correct but a native would never say this. We would say something like "I respectfully disagree", or "With respect, I disagree"

It would be a rare thing to say, but it is more subtle than saying something like ‘I disagree’. ‘Divergent’ means ‘different’ or ‘becoming different’. Especially with the word ‘respectfully’ in your statement it suggests the latter meaning. Perhaps you agreed with some of the premises of the other commenter, but came to a different conclusion. (I.e. you diverged) For example, a good use of the word ‘diverge’ would be I agree with your analysis of the problem, but diverge when it comes to your proposed solution. It’s too simplistic to ever work.

I agree with Graham's comments. I want to add, however, that "I beg to differ" can be a bit aggressive. It is often used with quite a combative tone, to express that we not only disagree, but feel very strongly about our opinion. If you want to be polite, it's best to use the phrases Graham suggested. If you want to get into a heated discussion, "I beg to differ" is perfect.

Your sentence is grammatical, makes sense, courteous, and sounds like something a well educated person might say. I don't know how things are in your country, but here in the U.S. the opportunity for sensible discussion about controversial topics is becoming vanishingly small. Here you have to be careful over with whom you discuss such things. If I start a sentence with "I have a different opinion...", more often than not it serves as an invitation to be told that I am wrong. So often the person interprets this as opportunity for him to tell me, without sensible justification why, that his opinions are the correct ones. Why is this so? Because there is so much enmity, so little clear thinking, and people simply do not understand one another. So, in sum, your words are beautiful and I envy you if it actually makes sense for you to use them. I personally find it hopeless to argue. A better strategy, in my opinion, is to invite the ignorant to explain their misguided opinions to you. When they say something absurd simply nod your head and say "how interesting, please tell me more". The only hope is that be being forced to hear themselves say ridiculous things that someday they might come to understand how misguided their opinions are.

In actual fact, you don't have to say that you disagree. You can simply say what you disagree on. "Oh, I'd always thought the moon wasn't made of green cheese." That is gentle and polite.