The Protestant Reformation is based on, among other things, "sola scriptura"--that the Bible alone is the basis for all doctrine and practice. There are probably two essential points in this matter.
The Priesthood of All Believers
First, a common doctrine of Protestants is the priesthood of all believers. It is believed that there is no biblical precedent in the New Covenant for a priestly class that is above the laity. Rather, the New Testament appears to affirm that all believers are priests.
You also, [God's elect, scattered throughout Pontius, Galatia, Cappadocia, Asia and Bythinia] as living stones, are being built up as a spiritual house for a holy priesthood, to offer up spiritual sacrifices acceptable to God through Jesus Christ. 1 Peter 2:5 NASB
But you are a chosen race, a royal priesthood, a holy nation, a people for God’s own possession, so that you may proclaim the excellencies of Him who has called you out of darkness into His marvelous light; 1 Peter 2:9 NASB
No Precedent for Hierarchical Confession
Secondly, there is no biblical precedent, so it is believed, for the practice of hierarchical confession--that the laity would be obliged to confess all their sins to a man of the clergy.
Perhaps the closest verse that supports confession to men is James 5:16:
Therefore, confess your sins to one another, and pray for one another so that you may be healed. The effective prayer of a righteous man can accomplish much. James 5:16 NASB
However, this does not support the idea of hierarchical confession, but merely confession of sins to one another. It would seem appropriate, then, for a layman, after confessing his sins to a priest, to then switch places with the priest and have the priest confess his sins to the layman.
Even in the Old Testament, David confessed his sin to God in Psalm 32.
I acknowledged my sin to You, And my iniquity I did not hide; I said, “I will confess my transgressions to the Lord”; And You forgave the guilt of my sin. Selah. Psalm 32:5 NASB
So, based on the conclusion that there is no biblical foundation for the priestly class of clergy, but rather an affirmation of the priesthood of all believers, and also based on the lack of biblical foundation for hierarchical confession, the vast majority of Protestant churches do not practice confession in the Catholic manner.