Council leader Julian Brazil said expectant mothers should not have to make journeys of up to two hours on rural roads to access maternity care.
He said: “Consultants and doctors should be driving to Barnstaple, not mothers-to-be and their loved ones travelling up to two hours.”
The trust has previously said patient safety remained its “overriding priority”.
It said, while the suspension continued, “all current support arrangements remain in place”.
Additional clinical, financial and travel support had been put in place to support women during their pregnancy, including extra appointments, accommodation expenses and how to access free taxis, the trust said.
Women are advised to contact maternity triage on 01392 406616 early if they think they may be in labour or have urgent concerns.
A Department of Health and Social Care spokesperson said: “This government inherited a broken NHS with the hospital estate left to crumble, and maternity and neonatal services in need of major reform.”
They added that the government had “committed £271m through the Estates Safety Fund into schemes related to improving the safety of maternity and neonatal estates”.
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