Electronic – STM32F4 USART Receiver Pin Configuring

Which STM32 processor are you using? The behavior you describe is typical of many processors, where it's necessary to explicitly connect peripheral drivers to output pins, but where they are implicitly connected to inputs (since such connections are generally inherently harmless). On the other hand, when I was using an STM32L151, I had trouble getting the UART to work until I selected the pin's "alternate function" as the UART. Are you still setting the AFR for that pin to select UART? I wouldn't be surprised if the pin mode can be freely selected as either "input/float" or "alternate function", since the UART wouldn't be driving that pin anyhow, but that it was nonetheless necessary to set the AFR of that pin to the UART, so that the UART would know from which pin it should accept input.

A topology I rather like, but which I haven't seen elsewhere for some reason (even though it seems like the obvious "right" approach) is found in some Microchip parts, where each pin's output has a mux which selects the peripheral which should feed it, but each peripheral's input has a mux which selects one pin to feed it. This allows one peripheral to drive multiple output pins, and allows multiple peripherals to be driven by one input pin. I recognize that in a large part a full crossbar arrangement might be impractical, but nonetheless selecting peripheral inputs that way would seem to have a number of advantages, such as being able to feed a pin both to a UART and to a timer event trigger (thus getting a precise timestamp of when a character arrives, even if the CPU can't service it instantly).

Note:
This answer was originally written before we knew anything about the receiver, like its 1.8V power supply. Telaclavo's answer is good. As a more general answer the PNP/MOSFET solution remains; you don't want to power parts of your circuit from a microcontroller's I/O pin.

You don't want to do that! You'd configure the pin as output and make it high to provide power to the GPS receiver, but microcontrollers I/Os can only supply limited current, 25mA for the STM32F205xx (Section 6.2 page 72 of the datasheet), which will be too low for powering your (and any other) GPS receiver (34 to 38 mA, as stated in the datasheet)

Use the I/O pin to drive a PNP transistor which will supply the required current.

Note that using a PNP inverses your logic: a logic low will turn the receiver on.

I would not use an NPN for this. In common emitter it would mean that the receiver's ground is a few hundred mV above ground, and a circuit should have one single ground which is the same for every component. In common collector you would lose too much of your 3.3V power supply.

edit
Wouter would use a MOSFET instead of a BJT, and that's a good alternative. Just make sure you choose a logic-level FET, which will give you enough current at a \$V_{GS}\$ of -3.3V. The Rohm RZE002P02 is a suitable type. It will also have a lower voltage drop if your receiver needs less than about 200mA.

edit 2 (re clabacchio's addition of a datasheet)
This device operates at 1.8V, the STM32 at 3.3V. You can use an LDO with an enable input and control that from your microcontroller. No transistor needed. (Thanks for the suggestion, markrages.) You'll also need level shifters for the data.