You don't need a 1V supply. You need something relatively more complicated than than.
While how much effort it'll take depends on what kind of accuracy you're looking for, let me try to explain whats going on and how you can get your sensor to work.
First, the humidity sensor doesn't work with a DC power supply. It needs a 1KHz sine wave with 0 DC bias and an RMS voltage of about 1V. This translates to a sine wave between about +1.414V and -1.414V. You need a bipolar sine for this, as any DC bias is likely to make the sensor readings incorrect and may even damage it.
Second, what you're measuring isn't the resistance, its the impedance. Impedance is a combination of L, C, and R components of the sensor. You may see this in example circuits called a DUT - Device Under Test. I would guess the sensor uses a change in epsilon (permittivity) and / or mu (permeability) caused by the introduction of a dielectric - moisture. This means that you're actually looking for changes in capacitance and / impedance. (Since water is non-magnetic as far as I remember, its most likely to do only with capacitance, but that's not the point and isn't important).
Further, it looks like only the magnitude of the impedance is important, and not the phase offset. This is probably the only saving grace.
With this 'background', lets see what you'd need for using this sensor.
The sine wave (1V RMS) is best to use as a sine wave because that means the signal has what we call spectral purity. It is made up of only one component in the frequency domain. If the wave is not a pure sine wave, your measurements will start becoming incorrect and the greater the impurity, the less the relevance of their calibration table to whatever data you produce. A square wave is probably a very bad idea.
The best way to generate a sine wave is using a sine wave generator IC like a XR2207, which is easy to use but unfortunately hard to get since its been obsoleted. There may be other options available similar to it. Alternatively, you could use a DDS (Direct Digital Synthesis) IC like Analog Devices AD9833 which can generate high purity sine waves. In this case, since the amplitude and frequency is fixed, it may be simpler and make more sense to build a sine wave oscillator like the Wien Bridge Oscillator. You would have to tune it to a frequency close to 1KHz and an ampltitude close to 1V RMS or 2.818V Pk-Pk, and ensure that there isn't a DC bias in whatever signal you generate. A capacitor in series with the signal can block the DC component. This is called AC Coupling and the capacitor is sometimes called a DC blocking capacitor.
A cheap and easy solution that you can try is to generate a 1KHz square wave from the Arduino, and pass it through one or more low pass filters tuned to about 1.2 or 1.3 KHz. This could block the higher frequency components and turn the square into something like a sine. Note that this won't be ideal and will definitely cause issues with accuracy. As I said, depends on how accurate you want it. In this case, also, you will need to have a DC blocking capacitor in series to get rid of the 2.5V DC bias. The other side of the capacitor, where the sensor is, may need to be biased to ground using a high resistance.
This part lets you excite the sensor. When it comes to measuring what you get, you can't direct send it into an ADC. You first have to convert the AC signal into DC measureables. By AC signal, I mean in this case that you want the AC amplitude (RMS) and the AC current. The AC amplitude you can measure by using a full bridge rectifier and a low pass filter. This would let you send a DC signal into an ADC. There are more accurate circuits for doing this, a search for RMS to DC convertor should let you find one.
To measure AC current, you'd include an additional resistance in series with the sensor and measure the RMS voltage across it using the same technique as above.
Using the AC current and voltage, and the AC excitation voltage (1V RMS), you can calculate the impedance and therefore the humidity based on the datasheet's calibration table.