Using a Single Supply OpAmp in inverting mode
An opamp can only generate output voltages within the range of its power rails.

If you supply the +ve and -ve power rails of an opamp with 5V and 0V (ground) respectively, you limit the linear range of the opamp output to 0V to 5V (± the rail offset of the particular opamp you're using). This means that any output voltage created by the opamp must be within 0-5V, otherwise you will hit the saturation range and "clip" your input signal.

In an inverting configuration, this means that if your input voltage is positive, you will always be in the saturation range, because any positive input voltage will attempt to create a negative output voltage, but your linear range is only 0-5V, so you saturate the opamp and get 0V out.

As you mention that "the input signal would never go negative", your proposed circuit would not work: your output would always be 0V because you would always be within the negative saturation range of the opamp, regardless of its gain.

As a side note, I should mention that the use of the terms "GND" and "Vcc" in relation to opamp supplies is largely erroneous, since the more appropriate terms are positive and negative rails (+ve and -ve). Voltage is a relative measurement, so tying the +ve pin to your circuit ground (0V) and the -ve pin to a -10V supply is absolutely acceptable, assuming you want to generate output voltages between 0V and -10V.