The problem that you have is that while he may have been complying with Florida law in giving a Notice of Intent to Lien, Georgia apparently has no corresponding law. He did work on your property in Georgia. All that he has to do to perfect a mechanic's lien against your Georgia property is comply with Georgia law. The fact that he is a Florida contractor probably has no impact in whether he has the right to be paid on a project he did in Georgia (I say probably because I don't know whether the work he did required a license and whether the license is a prerequisite to payment--it is in some states).
I am not a Georgia lawyer, however a quick review of the statute appears to say that he has 90 days from his last work on your project to file a mechanic's lien. That will have to follow Georgia law and be filed in your county. One route you can take is to wait and see. Many contractors send notices of intent with the hope that you will just pay under the threat. If you do decide to go that route, hire a local construction lawyer to help you. Check Avvo for an experienced lawyer that has a significant part of their practice in construction.
Although if he decided to sue you he COULD sue you in Florida, that is really stretching it. He should have to file a suit in your county. If he sues you in Florida, you will have to find a Florida lawyer (again, I recommend construction in that location), to defend the suit and, move it to your town. If he sues you in Georgia, unless the amount is negligible, you should hire a local lawyer to handle the case. The same Georgia lawyer I recommended you find is the one you should consider for the case.
It may be helpful to hire the Georgia construction lawyer now to have him or her review your case and send a "go away" letter to the contractor as preventative medicine.
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You should hire a construction law attorney in FL and a real estate attorney in GA. I don't see how the FL law gives rights to property in GA, but you should have local experts advise you and represent you in this matter. Good luck.
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Mechanics lien across state lines? I don't think so. If he sues you, gets a judgment, has it domesticated in GA, gets a writ, then he can lien it. But ask a real estate pro.
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This situation is actually rather common. The Florida contractor is trying to do his own legal work and does not understand that lien laws are state-specific. However, he may very well figure that out. If he looks into how to file a lien in Georgia, he should also be able to find a lien filing service that knows how to do it correctly.
The other answers are correct that he can't file a lien in Florida; that Georgia procedure does not follow the Florida statute; that a lien would have to be filed within 90 days of last work; that it must be filed in your county. He is likely not going to be able to sue you in Florida unless you did something that created consent to being sued down there.
Now, as to the money issue: You say you didn't pay full price, so apparently you made partial payment. Then you hired someone else to correct and finish the job. The main question is: Did the hiring of the new contractor cost you more than the balance left on the first contract? You are entitled to the "benefit of the bargain." That means getting a certain scope of work done for, say, "$50,000." If you paid the Florida contractor $30,000 and then paid the second contractor $25,000 to FINISH AND CORRECT THE SAME SCOPE OF WORK, then your project cost you $55,000, and you have a claim (or counterclaim) against the Florida contractor for net $5,000.
BUT: Under the same principle, if you held back $20,000 and then the next guy was able to complete the job for only $15,000, then you actually owe the Florida contractor $5,000 more. That is, he brought your project within $15,000 of completion, so he's entitled to $35,000. I deal with these types of cases all the time and people get confused on how the rule of damages works. If this situation needs to be pursued further, then yes, you should consult a local construction attorney.