An appeal of a jury's finding of not guilty is not the equivalent of being re-charged. The double-jeopardy provisions of the US Constitution do not come into play in an appeal - it is not double jeopardy because it is merely a continuation of the original proceeding. In truth, the DA's office could appeal the verdict if they want to. However in order to appeal successfully they would have to show the Appeals court an error in law (e.g., the judge's instructions to the jury were flawed, a piece of evidence was admitted that should not have been, a piece of evidence was excluded that should have been admitted).
In Canada the Crown Attorney could also succeed on appeal if they could prove the jury's verdict was so unreasonable that no reasonable jury, presented with the same evidence, could possibly have reached that conclusion. It is a very high standard and a very rare day when an appellate court will overturn a jury's verdict on that basis. The jury is the one that heard all the evidence and had the best opportunity to assess the credibility of the witnesses by observing their demeanour on the stand - their verdict is entitled to a significant degree of deference by the appeals court. I imagine there is a similar standard in Florida but never having practiced there I would not be inclined to speculate - Aardvark would know better what the standard would be to over turn a jury verdict in the US.
I am sure the DA in this case is closely reviewing their options for appeal - it is standard practice anytime a case is lost.