fundamentally disagree
profoundly disagree

"I take it by your answer that you profoundly disagree?"

Have to say that I disagreed profoundly with your tentative response to the question of a written constitution.

Again, what if the ruling circles of a society profoundly disagreed?

Living in reality means that people's impressions are important even if we profoundly disagree with them.

"But I profoundly disagree with the direction that John wants to take the country."

I disagree profoundly with its statement that tobacco farming is no different from any other agricultural activity.

However, it also contains comments with which I profoundly disagree.

But others profoundly disagree about whether it is good for democracy, or even constitutional, to try to put the brakes on such spending.

Despite his calling, he profoundly disagreed with capital punishment, saying that it served no useful purpose.