Need a comma?

As presently structured, your sentence does not require a comma. However, at 25 words long and without a comma, your sentence is considered too long for comfortable reading. Punctuation, such as commas, that breaks a sentences into parts helps make long sentences more readable.

Now that doesn't mean that you should just a comma here or there.  You may wish to restructure your sentence so that a comma is required.

"Believing they were going to spend more time in front of the computer and less with the patient, some of the doctors and nurses objected."

Hope this helps.