What many of the opponents fail to mention is that after paul gave that spiritual directive to the Jailer, it immediately states after that they spake unto them the word of the Lord, notice Acts 16:31-32
31And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
32And they spake unto him the word of the Lord, and to all that were in his house.
Now the word of the Lord is that which can also be classified as the things of the Spirit of God. They spake unto him : the things of God" !
Now if thats true, which none should deny, then paul says that the Natural Man receiveth not the things of the spirit of God. 1 Cor 2:14
14But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.
Now, I submit, that if the Jailor was in a Natural state or condition, and not born of the Spirit, He could not and would not have received and embraced the The Things of the Spirit of God, which the Gospel is.
Now did Paul mean the Gospel when he made this statement ? Yes, because context demands it, for in vs 1-2 of the Chapter demands that:
1And I, brethren, when I came to you, came not with excellency of speech or of wisdom, declaring unto you the testimony of God.
2For I determined not to know any thing among you, save Jesus Christ, and him crucified.
The Testimony of God , which is the Gospel, the preaching of Jesus Christ and Him crucified, its the Gospel.
So when one believes on the Lord it means they receive the Testimony of God as per 2 Thess 1:10
When he shall come to be glorified in his saints, and to be admired in all them that believe (because our testimony among you was believed) in that day.
Also, paul states in 1 Cor 2:14 that the things of the Spirit of God are foolishness to the Natural Man.
Paul says the same thing about the Preaching of the Cross to those that perish, it is foolishness 1 Cor 1:18
18For the preaching of the cross is to them that perish foolishness; but unto us which are saved it is the power of God.
The Natural Men are the ones perishing.
So in light of this another clear testimony from the word of God, we perceive, for the Jailor to have believed on the Lord by way of embracing the word of the Lord, which are Things of the Spirit of God, He could not have been in a Natural state, but Spiritual, that he could have discerned the things of God in them 1 Cor 2:14
Paul calls the Testimony of the Gospel to the Corinth Spiritual Things 1 Cor 9:11
If we have sown unto you spiritual things, is it a great thing if we shall reap your carnal things?
14But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.
To deny this, is to reject clear scripture testimony !