The title is pretty much my entire question. If I order a 240V lamp from Europe that takes removable light bulbs, can I simply use a 120V light bulb? I understand that I would change the plug for a plug-in light, but this particular question is for a hard-wired lamp.
Hard to say without details. If it is an incandescent lamp with a replaceable screw-in bulb, and the socket fits a bulb made for here and 120V, then yes, it shouldn't matter.
But again, the devil is in the details. If we are discussing screw-in base bulbs, Europeans mainly use E14 and E27 bases. E14 is equivalent to what we in North America call a "medium" base, which is not very common (specialty lamps like aquarium bulbs). We use "candelabra" bases for small bulbs, which is equivalent to an E11, but that is uncommon in Europe. Larger bulbs in Europe are E27 bases. The E number has to do with the width of the base; E27 is 27mm wide. Here in NA we use what's called an "Edison base" and it is 26mm wide, what the EU would call an E26. So it's possible, because of tolerances, for an E27 bulb to screw into an Edison (E26) socket, meaning you can use an EU light bulb in NA fixtures (however the voltage is usually wrong). But going the other way; an Edison (E26) bulb screwed into an EU E27 socket, results in the bulb being loose in the threads and possibly getting jammed, then difficult to remove.
If the lamp has a CFL or LED, then it would depend on whether the "ballast" for the CFL or the "driver" for the LED is able to accept the different voltage. A whole different kettle of fish.
Answer is that probably you can, but you really shouldn't.
Cables, switches and contacts inside the lamp are rated for specific voltage and current. You can go lower on both, but not higher.
Ampers go up if you want the same power with lower voltage. For example , if your lamp accepts 60 watt light bulb at 240V, then cables etc are rated for 0.25A. Similar 60 watt on 120V would require 0.5A, something cables in the lamp cannot take. You are limited to 30 watt bulbs then. Make sure to replace stickers to prevent mistakes.
Of course, if there are any active components, like electronic switches for touch turning on and off, or brightness controls, these may stop working when undervolted.
If you are buying something exclusive enough you have to import it from another continent, good chance is that you are buying from a small manufacturer. If that's the case, ask them if you can do this, what would be the limits and if they could make a lower voltage version for you.
If you still want to do it yourself, open the lamp to see if there are any active components and what amper limits are there on switches and bulb holders. If you can, you may replace wires for much thicker ones, to be sure. If you will get stuck, or need a specific electrical advice, you should not guess but ask on a sister site Electrical Engineering.
Still, I'd rather recommend buying lamp that simply works with the electrical installation you have.
Bulb sizes are nicely discussed in another answer. Don't screw in a bulb that would be lose.
I bought a really unique ceiling lamp in Dubai a few years ago, and it was set up with E-14 Euro style lamps. They're (barely) available here in Canada in incandescent but short-lived and rather pricey.
Found some nice 4W 120VAC (35 or 40W equivalent) LED 'filament'-look bulbs that fit it and were dimmable (with a proper LED dimmer), bought the set with one spare and it's been working since. Another option was to buy adapters from E-14 to the tiny E-12 base, but I chose not to do that.
The fixture is not approved by CSA or UL but it's properly grounded and running at half the design voltage (and much less heating and current), and is all metal/glass, so it presents negligible risk, in my opinion, and I've seen much worse stuff for sale at big-box stores.
If dealing with 1" Medium base Edison sockets when manufactured IEC standards, E26 and E27 actually have the same diameter specifications of 26.05 mm (min) to 26.45 mm (max).
The notable difference is that bulbs intended to operate at 120v have a thinner insulator between the tip and the shell, the minimum insulator height for E26 lamps bulbs is 3.25mm and for E27 is 5mm.
So putting a E26 bulb in a E27 socket at 240v could allow an increased arcing risk, but at 120v should not be a problem.
Watts are little more confusing, watts are heat, 3.41 btu's per watt/hour. But is the rating just a limit on the heat the sockets or shades can withstand, or is it a derived number setting the current limit of tiny components? An extreme hypothetical example could be they could make a fixture designed for six 100w bulbs, at 240v that would be less than 3A, so they only need to design internal components for approved for 3A. But if you were to plug in 100W 120v lamps at 120v you would be up to 5A. But in reality you are probably dealing with 3 or less 60W bulbs at 120v, you are only dealing maximum current of 1.5A. Do I think that's an issue? Probably not, but I can't prove it. I could also be concerned I've seen European plugs with built in fuses, the only place I seen plugs like that in the US were on 1970's Christmas lights, there might be some hesitation on my part cutting off a fuse and putting on an unfused plug, into a time delayed 15A circuit without blow characteristics required by lamp maker or approving agency.