Justification is clearly an act of God's grace, because what we deserve from what we have earned—from what we have done, the conduct of our lives—is death. There is none righteous, no not one (Romans 3:10; Psalm 14:1). Since justification, then, cannot be claimed as a right because we have sinned, it must be received as a gift. That fact that it is given makes it an act of grace.
It is not our hanging on to Christ (that is, the keeping of the law) that saves us, but rather Christ hanging on to us. That is, it is not what we do, but it is what He does continuously as acts of grace that saves us, because we deserve death. If we can earn salvation through law-keeping, Paul is saying in verse 21, "then Christ died in vain." If we can earn salvation through law-keeping, then Christ's sinless life and agonizing death were not necessary, because we can do it ourselves.
Justification is not vindication or exoneration. Both of those words connote that a person was right all along, but the true facts were hidden from those who were doing the judging. In some cases with men, vindication is possible because people are judged unrighteously. Their judges are not using righteous judgment.
But God never judges unrighteously! He knows all the facts. He knows our heart. He knows everything about us in every situation that we have ever been in, so He cannot vindicate us because we are not clear of guilt. He cannot exonerate us because we are not innocent. Justification is more than that. It is setting us right or calling us righteous though righteousness does not exist in us.