Launching and assembling the ISS in its current design could not have been accomplished without the STS (Shuttle system), but that is largely because the ISS was designed with the STS in mind as its launch vehicle. The ISS could be designed to be assembled in orbit using a large number of spacewalks because designers knew they would have the shuttle available to host EVAs. While the STS boasts a payload capacity to LEO of 27,500 kg, the largest ISS components it delivered were all under 16,000 kg. The largest ISS components at 19,000 to 20,000kg were delivered by Russian Proton rockets.
Wikipedia has an excellent article on the Assembly of the International Space Station which describes module sizes, delivery methods, and the number of components and related spacewalks.
However a space station of a different design could have been launched and assembled by different rocket families. In fact, two other space stations were.
Skylab was launched in 1973 on a modified Saturn V booster. Skylab weighed in at 77,000 kg, four times the weight of the largest ISS modules, and had an internal volume of 12,417 cubic feet, slightly more than 1/3 of the ISS' current volume. So in the broadest terms, a space station with similar stats to the ISS could have been launched and assembled using three Skylab-class modules. Skylab's crew, launched separately in an Apollo module, transferred to Skylab and then used it to host EVAs, most notably to repair the lab, as it was damaged seriously during launch.
In 1986 the Soviet Union used a Proton heavy-lift rocket to launch the first component of their space station "Mir." Mir would eventually be composed of seven pressurized and several unpressurized modules with a total pressurized volume of 350 m^3 (12,360 cu ft). Although eventually decommissioned as Russia joined the US as an international partner on the ISS, Mir was a very capable space station and was assembled in orbit and crewed using Soyuz capsules without the Shuttle-like capabilities of the Canadarm and extensive EVA support.
In summary, the ISS components could have been (and frequently were) launched on non-STS rockets, but assembly in its current design required the support of the Shuttle. Other, earlier space stations were launched by other systems and assembled by different methods. Given the lessons of Skylab and Mir, it stands to reason that a space station with capabilities similar to those of the ISS could have been built, launched, and assembled without the STS. "Easier" is difficult to quantify. If we had stuck with the Saturn V-based launcher and Skylab-class modules, we would have faced far fewer assembly EVAs but with much larger, more massive modules. In lieu of the Shuttle's arm and EVA support we would have had to develop and qualify a docking system that allowed an 85-ton module to rendezvous and dock with another 85-ton module as "easily" as a 32-ton Apollo CSM docked with its LEM.