As lots of people have said no its not proven and it will remain a debating point for some time due to all the reasons everyone has outlined already. However it would be an extremely foolish public health official that didn't issue that warning in the light of the epidemiological evidence we do have. Better to be wrong later if it turns out not to be such a big problem as the other way round. 

We are a veterinary institute and have some painfully gained first hand experience of a very similar situation with the Schmallenberg Virus (SBV) outbreak a couple of years ago. Its also an insect borne virus (which makes spread near impossible to stop), it causes mild or no disease in adults. At the time it was identified nobody was sure if it was really anything to worry about or not. However we did know that other genetically very similar viruses do cross the placenta and replicate preferentially in fetal neurological tissue resulting in birth defects when the resultant calf/lamb is born several months later so we were watching for this and that is precisely what happened the following lambing season. Some of these newborns were deformed, some were dead, you could detect virus in some but not others, you could detect serological responses by the fetus in some but not others  and that is entirely to be expected(each fetus and pregnancy is going to handle the virus slightly differently). When people did experimental infections only a small percentage of deformities/infected fetuses could be induced so the "window" of it getting across the placenta (even with a virus we know does this) is small.  

The reason you don't see such outbreaks in areas where the virus is already present (in the case of SBV like Zika this is probably somewhere in central Africa) is that most susceptible individuals are infected in their first couple of years of life and become immune well before they reach reproductive age. The big problem is with infections at a specific stage of pregnancy where the virus can get across the placenta, important organs are developing and the fetus isn't able to mount it's own immune response (fetuses close to being borne can fight off the infection themselves quite often, ones at earlier stages don't have the tissues that the virus replicates in yet and how the mother transfers things to the fetus changes over the course of a pregnancy). So you really only see mass outbreaks of deformities with SBV/Akabane (and presumably with Zika if this all pans out) when the virus is spreading into a new area or if climate conditions mean the insect (and therefore the virus) have been absent for a while and there are now lots of susceptible pregnancies again. 

In the case of Zika we do know that flaviviridae (of which Zika is one) can cause precisely this range of problems for fetuses because there are veterinary ones that do this. The  cattle one is  called Bovine Virus Diarrhea Virus (usually known as BVDV as it less of a mouthful) that causes neurological deficits in calves infected in-utero. It is also persistently excreted in semen from infected entire males and sexual transmission can occur.  This particular virus also has a peculiar problem in that calves that are infected  in-utero while their immune system is still at the stage where it is working out what is "self" and what is "non-self" will not recognize the virus as a virus and will never eliminate it - they will excrete it for the rest of their lives. These animals are known as "persistently infected"  and are a major problem in trying to eliminate BVDV from infected herds. 

In terms of animal models - the problem is that the placenta is wildly different between different mammals - there are some old papers where people have infected mice with zika but how good a model this would be for human cross placental transmission is very debatable. You are really going to have to use primates (which are not the easiest things to work with experimentally for a whole variety of reasons) 

So in terms of the original question no, we haven't fulfilled koch's postulates that Zika causes microencephaly in human fetuses (and are never likely to) . However given what we do know about BVDV  and similar viruses and the current epidemiology it is right to be raising this concern at present. The good news is that if Zika follows the same sort of epidemiological pattern as SBV the number of cases in fetuses in future years should drop back to "normal" (whatever that is) and the advice to delay pregnancy till after you've been infected (or vaccinated if one becomes available) is a sensible one.