According to LabCorp spokesman Mike Geller, because those states have bans on the use of home testing kits:
The test's website says it will not be available in New York, New
Jersey, Maryland and Rhode Island. Those states are excluded because
they have regulations against patients initiating their own lab tests,
Geller said. New York and New Jersey are the two states with the
highest number of coronavirus cases and deaths.
This 2017 TechnologyReview article helps explains why, at least for NY and MD:
Orig3n is one of a slew of new companies that promise to provide
personalized health information on your skin, athletic ability, diet,
intelligence, and other traits based on your genetic makeup. Known as
direct-to-consumer genetic tests, they are available at the click of a
mouse for $200 or less. You spit into a vial or swab your cheek and
send your saliva through the mail, no prescription sign-off from your
physician required. But two states—Maryland and New York—are holding
out against these tests.
Maryland state law allows only doctors or other authorized health-care
practitioners to order lab tests, with a few exceptions. Consumers can
directly purchase tests that have been approved by the U.S. Food and
Drug Administration as “over-the-counter” devices. Orig3n has not
received that designation.
Maryland also doesn’t allow lab tests to be directly marketed to
consumers, only to physicians, hospitals, and laboratories.
This line has however been challenged by Paul Celli, public health administrator for clinical and forensic laboratories at the Maryland Department of Health Office of Health Care Quality, who wrote to the Cato Insitute:
Whatever may be the situation in the other three states, “Maryland is
not banning this Pixel at‐home collection device (it is not a test)
for use at select Labcorp testing locations.” A Maryland legislative
source points out that last year, state lawmakers approved and Gov.
Hogan signed SB 495, a measure aimed at liberalizing access to medical
testing by removing some of the restrictions in effect earlier. Mr.
Celli writes that even before that change, rather than bar use of this
particular test the state “probably would have exercised enforcement
discretion in such cases where the company appears to be providing
services pursuant to a physician or other authorized provider order
for the test.”
So while things are a little inconclusive with regard to Maryland, nevertheless this appears to be at least LabCorp's published reasoning for not extending the test to these four states.