The 24 crew, mostly from Eastern Europe and Asia, are still on board the Maersk Tigris, anchored off Bandar Abbas.
Maersk said it still did not know the reason why the Marshall Islands-flagged vessel was seized, and noted it had been in an international shipping lane.
But Iranian authorities noted that it had been in the country's waters.
Iranian media cited an official as saying the ship was seized based on a court order in connection with a complaint made by a private Iranian company about debts.
On Wednesday, Maersk said it had been in contact with Singapore-based Rickmers Shipmanagement, from whom it charters the Maersk Tigris, and was "pleased to learn that the crew is safe and under the circumstances in good spirits".
The crew includes a British national, but no Americans, as initial reports suggested.
"We are continuing our efforts to obtain more information about the Iranian authorities' seizure - in international waters - of Maersk Tigris. We are not able at this point to establish or confirm the reason behind the seizure," Maersk added.
"We remain in close dialogue with the Danish ministry of foreign affairs."
A spokesman for Rickmers Shipmanagement said the vessel had been carrying commercial goods and no "special cargo" such as military equipment.
The incident has prompted the US Navy to dispatch a destroyer and a reconnaissance plane to the area to "monitor" the situation.
A general distress call sent out by the Maersk Tigris on Tuesday morning was picked up by US Naval Forces Central Command (NAVCENT).
A Pentagon spokesman said Iranian Revolutionary Guards Navy patrol vessels had approached the ship while it was moving through Iranian territorial waters, but inside an area that under international agreement is open to foreign ships making an "innocent passage".
The patrol vessels ordered the container ship to go deeper into Iranian waters, but the master refused. Only when they fired warning shots across its bow, did he comply.
The Pentagon spokesman called the actions of the patrol vessels "inappropriate".

The reforms, ranging from the setting up of small businesses to reducing bureaucracy, had already been agreed by the ruling Communist party.
Mr Castro also hinted that changes to the country's controversial travel and emigration rules could be introduced.
He said those attempting to undermine the reforms could face prosecution.
The BBC's Michael Voss in Havana says one of the biggest obstacles Mr Castro now faces is resistance from party bureaucrats who face losing their job under the changes.
But the president warned lawmakers that "bureaucratic resistance is useless".
Mr Castro told parliament that the time had come for social and political reforms to be introduced. "Let's clean our heads of all sorts of nonsense," he said.
The measures the assembly was discussing in its twice-yearly session include cutting more than one million state jobs in a move to reduce Cuba's vast bureaucracy and reducing the state's role in areas such as agriculture, retail and construction.
Small private businesses will be encouraged to step in to fill the space, while state subsidies for goods and service will be phased out.
Mr Castro also said the government was "working to orchestrate the modernisation" of the country's migration policies.
Human rights groups have long criticised the requirement for Cubans to get government permission to travel abroad - a remnant of the days when Cubans emigrated for political rather than economic reasons.
Mr Castro said the rules had "played a role in their time and endured unnecessarily".

The Red Rose, on the brink of defeat by Surrey this week, have lost two and drawn five of their previous seven County Championship games.
Lancashire won three of their opening five Division One games this season, but last triumphed in May.
"We are looking behind us now," Giles told BBC Radio Lancashire.
"It would be nice if results went our way elsewhere around the country, but ultimately we can only do what we do and we need to play better cricket for the last three matches.
"We're getting a bit of a wake-up call here as there is a lot of cricket to be played and it is a very tight division.
"At the moment we're far from safe so we've got to play better cricket."
Lancashire sit fourth in the table after 12 County Championship matches, 17 points above second-from-bottom Hampshire in the relegation places.
The crew of a container ship seized by Iran in the Strait of Hormuz on Tuesday are safe and "in good spirits", Danish shipping company Maersk says.
[NEXT_CONCEPT]
Cuba's National Assembly has given its backing to President Raul Castro's plans to reform the country's stagnating economy.
[NEXT_CONCEPT]
Lancashire cricket director Ashley Giles is concerned their end-of-season form could see them dragged into a relegation fight.