The adversarial system uses specific laws, precedents, and legal rules to determine who wins. It allows lawyers to take the truth and spin it into the picture that is the most beneficial to their client. They can use loopholes in the law to keep evidence from being allowed to cause their guilty clients to be punished for what they have done. Once all the talking is done, it is up to twelve people to decide whose lawyer did a better job of convincing them to believe them.
The civil system uses general ideas and broad concepts to form the framework for taking the evidence at hand and attempting to determine what the truth is. When the truth is revealed, lawyers do not try to hide it or escape from it, even if they do not like the results. A major disadvantage of this system is that those twelve people from the common law system are only used in major criminal cases, so when the truth is unclear, only a couple people get to determine who is right, and three people can be wrong easier than twelve people in 100% agreement.
In my opinion from what I have learned, I believe that the civil law system has an advantage in finding the ideal of the judicial system - justice. In the adversarial system, the truth can be avoided, and once it is, there can be no justice.
As to the question of whether there are constitutional problems with applying civil laws in the US, the most obvious one is that a jury trial is guaranteed by the Bill of Rights to protect a defendant from being condemned by the voice of one person.