Density isn’t going to work easily, if at all.
Mass per unit volume. There isn’t an obvious physical property that any sort of sound measurement can make that is characterised by just this alone. Bouncing sound off an object might, with enough effort, manage to measure the admittance of an object. But this almost always needs a contact measurement. Similarly (and related) you might measure the speed of sound in an object, so long as you know its size. Assuming you can get a reflection of the far boundary, and the reflection is reasonably clear.
An ultrasonic sound wave travelling in air is either going to reflect off, be adsorbed, or travel into the object, or a mixture of the three. The dominant factor is the impedance difference between the object and the air (which relates back to the admittance). None of these are directly dependant on the density of the object. The speed of sound in the material is related to the density, but you need to know the elastic modulus of the object as well. Same for admittance/impedance. A bucket of water and a solid lump of plastic have about the same density, but wildly different acoustic properties.
If this is the use case, you can step back quite a way. You don’t want density. It is the wrong metric. An ultrasonic sensor with a solid bit of signals processing could probably provide a metric of surface characteristics that might provide guidance.
A wall is flat, and typically somewhat rough, to the point where the high frequency adsorption spectrum might be used to characterise it. Even a painted drywall surface adsorbs at very high frequencies, and any sort of porosity will have a characteristic curve. Metal objects will be different again. Cars are smooth and curved. Bushes will adsorb a lot and generally have a messy signature.
A single ultrasonic transceiver might provide a starting point. However cheap ones typically only work at a single frequency. You really want to sweep the frequency. Better would be a phased array that you can actively scan. However that requires increased size, as you need the elements to be spaced apart in order for the beam steering to work.
In the modern world, the same, but better, capability might be provided with LIDAR arrays, such as used in phones or robotic vacuum cleaners.
Would help to know the use case. The question seems to be assuming the technology.