I'm using a STM32f373x and with no battery. The product specification on page 77 says it should be connected to VDD. I have made a test with Vbat pin left open and the micro is working. Why I should connect Vbat to VDD and why it may work if it's open?
There is an excellent video by Dave Jones of the EEVBlog explaining why you can power almost every IC through almost every pin. This applies to your situation as well since you're supplying power through VDD. Vbat will be connected through the ESD protection diodes (which are present on virtually all pins on the IC) to the VDD pin and taking it's supply via this "detour".
It is just good practice not to leave the Vbat pin open but instead connecting it to VDD. Although leaving Vbat open works perfectly well as you have found, the circuits feeding from Vbat will operate at a lower supply voltage because they receive their supply voltage through the ESD diodes. In certain situations this could lead to unwanted effects. For example the minimum supply voltage at which the processor will work might be different.