I agree with @Mark C. If she has a certificate of divorce, the divorce is legitimate. She is truly divorced and is free to remarry, whatever the circumstances preceding her getting that certificate of divorce.
However, if he was unjustified in divorcing her, then by divorcing her unjustly he was sinning. The divorce is still real, it still happened, and she is still free. But he will answer to God for this sin at the judgment, in the same way that we will answer for any other sin.
Thinking this over further, it has the implication that government divorce papers actually are meaningful. It's easy to say "let's divorce on paper" for various practical hypothetical reasons, but won't consider it a real divorce... How many marriages will actually last that? Not many. Because the divorce on paper did actually mean something. A government divorce is a written divorce agreement, and if the husband's signature is on it it's a real divorce in God's eyes. Now, if the husband's signature is not on it, and it was granted without his involvement as sometimes occurs, it is invalid. But if he signed it, it's real. So much as we may disagree with government involvement in marriage, now that they are involved, be careful what you sign.
However, if he was unjustified in divorcing her, then by divorcing her unjustly he was sinning. The divorce is still real, it still happened, and she is still free. But he will answer to God for this sin at the judgment, in the same way that we will answer for any other sin.
Thinking this over further, it has the implication that government divorce papers actually are meaningful. It's easy to say "let's divorce on paper" for various practical hypothetical reasons, but won't consider it a real divorce... How many marriages will actually last that? Not many. Because the divorce on paper did actually mean something. A government divorce is a written divorce agreement, and if the husband's signature is on it it's a real divorce in God's eyes. Now, if the husband's signature is not on it, and it was granted without his involvement as sometimes occurs, it is invalid. But if he signed it, it's real. So much as we may disagree with government involvement in marriage, now that they are involved, be careful what you sign.