The following is the passage used to say that there is a spiritual act of worship:

Romans 12:1-2 I beseech you therefore, brethren, by the mercies of God, that ye present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service. And be not conformed to this world: but be ye transformed by the renewing of your mind, that ye may prove what is that good, and acceptable, and perfect, will of God. (KJV)

Notice that verse one does not say “spiritual service, or “spiritual act of worship”, as some versions have it. The Greek literally says, “I am inviting (parakaleo – to call near or invite) then, you brothers, through the pities of the God, to stand/proffer/be at hand, the bodies of you (your bodies) sacrifice (act or victim, literal or figurative), living, holy, fully agreeable, to the God, the logical ministration/latreia (or “divine” service) of you (your)….” And so on.

(From the Thayer and Smith Greek Lexicon, latreia: service rendered for hire – any service or ministration: the service of God – the service and worship of God according to the requirements of the Levitical law – to perform sacred services)

The word spiritual is not even there.

The passage draws its language from the old type of service under the law, after the flesh. But by the mercies of God, it is reasonable to present ourselves to Him – for we are wholly His.

Notice that it also says, “living”. It’s a new and living way we are walking in. Not the way of death, but life. We “surrender” to God’s way, the living way, and the only sacrifice we can logically bring is our living selves – our hearts and our hands and our feet. Lively stones of God’s building.

We are in the spirit if we are Christ’s. We are therefore spiritual, before we “do” anything else!

Romans 8:1 There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit.

Ephesians 5:8 For ye were sometimes darkness, but now are ye light in the Lord…

Romans 8:9-10 – But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness.

Those who are Christ’s are spiritual, the old man being dead.

We all may not be mature, but we are all spiritual because our very nature has been changed – we are a new creation. Everything we do is actually spiritual. God fully accepts our acts done by our living flesh, for we are accepted in Christ.

I like it put this way: Romans 12:3 – (The Message) I’m speaking to you out of deep gratitude for all that God has given me, and especially as I have responsibilities in relation to you. Living then, as every one of you does, in pure grace, it’s important that you not misinterpret yourselves as people who are bringing this goodness to God. No, God brings it all to you. The only accurate way to understand ourselves is by what God is and by what he does for us, not by what we are and what we do for him.

Here is another, grace-based, take on that passage.  Paul is inviting them, because of God’s grace given by His mercies (he just spent 11 chapters talking about grace) to be ready as flesh and blood human bodies, to serve in the temple made without hands (for elsewhere it is written that God is not served by human hands).

In other words, we serve in the body of Christ, we who are now living in newness of life, with him working through us. The rest of the passage is the result of what the grace-walk looks like. 

It isn’t true spirituality to do anything in order to be spiritual; it will only fail. True spirituality is recognizing and acknowledging that we are already spiritual – and then walking in it.