Worship, properly so called, requires something be "offered up" to the object of devotion. It could be anything from incense (as in the cult of Roman Emperor worship) to a human being... or even a divine Person (as in the worship of Almighty God).
Adoration, for instance, requires that the object of devotion be acknowledged to be the source of his/her own power, authority, beauty, etc.. So Almighty God may be adored but the BVM may not be adored because she is the not the source of her attributes.
And just where do we have an example of Peter or Paul or James or any of the disciples doing this??
We have no command from them to do this.
As far as Mary, there are catholics who build shrines to her....burn incense to her...pray for her to do things for them....sounds like worship by your definition.