Before you start experimenting with the AF Fine Tune feature, it would be worth eliminating some of the other variables.
As Roger says, the depth of field of FX cameras is inherently shallower than that of cropped-sensor models (when viewed under the same conditions), so you may need to mentally adjust for that.
Secondly, it's not really possible to be sure exactly which part of the flower the camera has chosen to focus on. With a relatively "deep" target at such a close distance, the AF system may choose one "petal" one time and a different one next time. Roger's suggestion of checking with a flatter target is a good one.
Lastly, you haven't said which AF settings you are using. The D700 has a comprehensive set of AF controls, and it usually takes some time to understand which combination works best for each type of subject.