What do you understand by spiritual worship? I base my answer on the Larger Catechism, such a delightful guideline I find there concerning true spiritual worship, which is demanded of man in the first and foremost commandment (both the Shema, and the first in the Decalogue).
Spiritual worshippers, those kind of people which God seeks for himself, can be today denominated as Christians. Non-Christians in no way can or are even willing to worship God according to his requirements. Philippians 3 says "we are the true worshippers, who worship God in Spirit and put no confidence in the flesh." "We are the true circumcision."
True worship begins in knowing and acknowledging God as the only true God, and our God and Father. Pharaoh found no reason to obey or worship, "I do not know him," he retorted haughtily.
Our worship should be in accordance to God's being and revelation of himself. Will-worship is condemned and unacceptable. It is called "strange fire." Many people may sound sincere and zealous but they are inventing ways of worshipping instead of submitting to God's directives. (When we approach a head of state, there are certain established policies to observe; how much more this applies to God, the king of kings!)
Christians worship best by thinking of God, meditating upon him, remembering and highly esteeming him. All our life is lived "unto him." Coram Deo. We are called to adore, choose, love, desire and fear him.
The three cardinal virtues of faith, hope and love are to be directed towards him. Thus we are to give all praise and thanks and yield all obedience and submission to him with all our being. In worshipping Him God is pleased for this is his due and we are to be sorrowful and displeased when we fail in this fundamental matter.
Humility and a spiritual frame of mind makes our worship possible. The mind set on the spirit is life and peace (Romans 8). Thus man find true fulfilment is being lost in wonder, love and praise of his Creator and Redeemer.