It look like the page you are referring to is Electromagnetic absorption by water and it contains quite a lot of useful explanation about the absorption spectrum of water in the gas, liquid, and solid phases. Some of the descriptions on that page may seem to be a little bit contradictory at first sight, but the situation is not really complicated (but the liquid is certainly more complicated than the gas!).
In the gas phase, the water molecules hardly interact with other molecules, and can be treated quantum-mechanically as isolated objects. They have discrete rotational and vibrational (and electronic) states. Low-frequency radiation (microwave and far-infrared) cause transitions between rotational states: radiation is absorbed, but only at specific frequencies, because the states are separated by well-defined energies. We usually call this a "rotational absorption spectrum". Higher frequencies can cause transitions to excited vibrational states and these are usually accompanied by a change in the rotational state as well. In general parlance, we associate the word "spectrum" with the observation of these well-defined frequencies at which the absorption takes place.
At higher gas pressures, collisions occur more often between molecules. The absorption lines in the spectrum are observed to get broader. Each molecule can no longer be described as being isolated. In the liquid phase, interactions with surrounding molecules are happening all the time. The rotational states are affected more dramatically than the vibrations: the hydrogen bonding network largely prevents the molecules from rotating freely as they did in the gas phase. The key sentence on that page is
In liquid water the rotational transitions are effectively quenched,
but absorption bands are affected by hydrogen bonding.
So, it is not possible to distinguish individual rotational transitions,
but the molecules still rotate, albeit under the influence of their
and absorption of microwave energy can still take place.
It just happens
over a broad range of frequencies, without any distinctive lines in the spectrum.
On that page, in the section headed "Microwave and radio waves" you can see the sentence starting
Liquid water has a broad absorption spectrum in the microwave region
and a diagram showing how the absorption (dielectric loss) depends on frequency.
This is what is used in microwave ovens to heat food.
So I think they just meant, by the phrase "liquid water has no rotational spectrum", to say that the distinctive absorption lines in the gas-phase rotational spectrum are not seen in the liquid phase:
the spectrum has become featureless.
The vibrational spectrum, by the way, is also affected by the neighbours in the liquid,
but not so dramatically.
The vibration frequencies are much higher, and the vibrations tend to be
somewhat decoupled from the surroundings: it is still possible to identify
the characteristic lines in the spectrum.
[EDIT following OP comments]
Here are a few more references which I hope will clarify what is going on.
The water rotation in the liquid is not "free rotation", but is significantly hindered by the hydrogen bond network. This description of the physics of the microwave oven clearly states that
Free and undisturbed rotations cannot occur in liquid water due to the
interactions with neighbouring molecules
and this page explains
The re-orientation process may be modeled using a "wait-and-switch"
process where the water molecule has to wait for a period of time
until favorable orientation of neighboring molecules occurs and then
the hydrogen bonds switch to the new molecule
Computer simulation of, say, a few hundred or a thousand water molecules can give a picture of what is
going on (subject to some approximations in the way the water is modelled),
but it requires powerful computers. You can search online for videos:
I found this one which highlights the way the rotation is strongly
influenced by the rearrangement of the hydrogen bond network.
So a simple picture of free rotation followed by heat transfer to the surroundings is an oversimplification. Instead, the process of microwave heating
needs to be thought of as the dielectric response of the liquid to the applied
field, and it involves a combination of modes of motion coupled together:
molecular rotation is certainly a large part of this, but it is nothing like
the free rotation seen in the gas phase.
As it happens, a recent paper by ND Afify and MB Sweatman,
J Chem Phys, 148, 024508 (2018) uses computer simulations to
model the effect in this way (as a dielectric response).
It is available in open access form here.