No. Hospital staff members didn't skate around Cook County Hospital in the early 1990s. This seems to have been a bit of whimsy, perhaps driven by the appearance of one of Chicago's great rollerskaters on the set.

We can effectively rule out this being actual practice at Cook County Hospital, since someone who'd know said otherwise. In Cook County ICU: 30 Years of Unforgettable Patients (2015), Cory Franklin (a longtime physician at Cook County Hospital and a technical advisor to The Fugitive) writes the movie was pretty realistic in its depiction of the hospital "[w]ith the exception of the rollerblading orderly" and that when the rollerblader came down the hallway during one of the takes, he was "struck by the incongruity." That take ended up being used, so now there's this rollerblader in the background of the scene prompting questions like yours.

The rollerblader was played by Greg Hollimon, who is credited as "Skating Orderly." Hollimon was, at the time, a Chicago improv comedian and veteran of the Players Workshop and Second City. Before that, however, in the 1970s and early 1980s, Hollimon had been part of Chicago's skating scene. Skating at that time probably merits its own question, but basically start with disco on skates at a very high level of proficiency, both at rinks (where contests were held) and on the street, and then it keeps evolving and changing. Hollimon was considered a leading member of the scene. Hollimon's skating skills were sometimes used later in his comedy career. For example, in his breakout role on Strangers with Candy (1999-2005), Hollimon occasionally skated.

I have never seen a direct explanation from Hollimon or anyone connected with the production why he skated through the scene. Possibly it is in a commentary track.

So we will see if the mods allow this, as my answer comes from me working in hospital standards and compliance (along with a passion for medical history), and also from the fact that I have worked in various hospitals for 30 years, including candy striping as a teenager.

Depicting staff on roller skates and rollerblades is actually common in medical shows. Dick Van Dyke played the eccentric Dr. Mark Sloan on Diagnosis Murder from 1993 - 2001, who was known to do his rounds on roller skates. E.R., Scrubs, and Grey's Anatomy all featured rollerblading staff members.

As The Fugitive takes place in the US, I will restrict this answer to US hospitals. First off, no, there has never been a time when hospitals implemented roller skaters or roller bladers as a means of transportation. By the 1980s and 1990s, large hospitals had adapted a pneumatic tube system for sending urgent samples, images, or charts throughout the hospital. Now that health records have moved online, they are use primarily for sending samples for lab testing.

However, hospitals can be so large that they consist of several buildings over several blocks. Some cities have connected hospitals with staff working between them. It is common for hospital systems to have vast underground tunnel systems used for transporting things. Hospitals also often are connected to universities, and many nursing and medical students train and work there, and are traveling all over campus. So roller skating and blading have certainly long been popular ways to get around these hospital campuses, as well as razor and electric scooters when introduced. Some hospitals do allow (or have allowed in the past) these kinds of devices in certain parts of the tunnels.

OSHA standards for health care workers don't actually prohibit wheeled footwear; what is important is that they have closed toed shoes that can protect them from both spilled fluids and falling sharp objects. Roller skates and rollerblades do meet those requirements. So it is up to the hospital's individual policy as to whether or not they would allow it.

Some hospitals ban using these sorts of things inside because it creates a hazard both for the wheeled person and the staff/samples/equipment around them. However, some do not have specific rules, often because they just never thought about it, and every now and again someone (usually a student/resident) will try out using skates or roller blades or a scooter inside in the hospital hallways. Usually, this action will then result in a rule being created to ban them. I've only been able to find a few cases of clinicians regularly wearing roller skates in their practice, and these have all been cases of staff in places like nursing homes or pediatrician offices, where it is for the purposes of putting patients at ease and injecting some silliness into a scary situation, not for practical purposes. Here is an example below:

https://www.nj.com/indulge/2015/05/new_jersey_rolls_through_the_office_blowing_bubbles.html

So use of roller skates and roller blades in hospitals, while not nonexistent, is far less frequent than TV would have us believe, and they are MOST likely to be banned in the kind of places tv most frequently depicts them: In crowded hallways in and around ER rooms, with lots of people rushing around, many of them carrying delicate equipment and samples.

One thing that has become more widely used among hospital staff are wheeled sneakers like Heelys. They don't go as fast and aren't as likely to spin out of control as rollerblades, but most importantly, they are subtle, and can easily convert to regular shoes. So it is very easy for people to just use them without it drawing any attention, especially if they are just using them when traveling down long sparsely or unoccupied corridors. However, I also think (but am not certain) that these wheeled shoes are actually more likely to be specifically banned in hospital policy precisely because people are much more likely to actually try and use them.