If we were able to "do" the entire law—in the letter and the spirit—we could then "live" by that means. Paul shows that it does not require faith to keep the law in the letter—anybody can compare an action against a list of dos and don'ts and see if the action is allowed. It requires much more to keep the law in the spirit perfectly. It requires a full measure of God's Spirit working within the person. But simply to abide by a law does not require any faith in a Savior, so if this life were just about strictly adhering to a list of requirements in their letter, Christ would have died in vain.
Romans 8:7 says that the carnal mind is not subject to the law of God—that is, it will not submit itself to God's law. But there is ample evidence that unconverted man can live according to regulations in a Pharisaic manner. Romans 7:14 adds more to the equation by showing that God's law is a spiritual law—there is an intent behind it, as well as the most direct application. This was what Christ was endeavoring to show in Matthew 5:20-48. So for us to be justified before God, we would have to completely fulfill the law—live according to the letter and the spirit. But that is a logical impossibility without means of the Holy Spirit.
This is why justification by faith is a necessity: We need God's Spirit to fulfill this spiritual law, but God will not give His Spirit to someone who does not willingly submit to Him and obey Him. This is why God would not allow Adam and Eve access to the Tree of Life after they had sinned, because He knew that their natures had become corrupt, and He was not willing that a corrupt being be given His Spirit—His power. A paradox results, and the only way out of the deadlock is for God to bring a person into alignment with Him by substituting the perfect life of His Son for ours in a legal action. Once that justification has taken place, then a measure of His Spirit can be given, and the person can begin to keep His law in both the letter and the intent.