"accept the existence" is correct and usable in written English.
For example, you could use it in a sentence like, "She had to accept the existence of her illness before she could begin to heal."
Ben-Gurion believed that the Arabs would never accept the existence of Israel except by force.
It is, instead, simply a rational decision not to accept the existence of God without evidence.
We as singers accept the existence of critics as part and parcel of our working lives.
Maybe we cannot or will not accept the existence of the kind of unblemished love he claimed to represent.
This sort of person "does not accept the existence of an objective fact separate from how he feels about it".
Good faith on both sides requires that they implicitly accept the existence of the other as a fact of life.
You meet in a restaurant because you accept the existence and presence of others sharing the same public space, many of them strangers to you.
Whitman's refusal to accept the existence of boundaries and limits on the body, as well as the mind, is the most radical statement ever of American individualism.
The central conflict between Israel and its Arab neighbors is their refusal to accept the existence of Israel and their obsessive desire to destroy the Jewish state.
The others in the 5-to-4 majority were willing to accept the existence of such a principle, but decided that the sentence did not violate it.
One food bank cited in the report says this formalised delay "seems to accept the existence of food banks as a safety net".
