Their rusty ship was towed to the Black Sea port of Constanta.
Country said to have received about 56,000 asylum-seekers since 2012.
"Ultimately, this is about basic decency" Obama wrote in a statement.
Bibihal Uzbeki's son and grandson had to carry her on their backs on the family's journey to Europe.
The government is blaming a pro-Rohingya armed group for the violence, but people fleeing the violence accuse government authorities.
Dale McIntier says it could be a year before he can reapply for a spousal visa if he's forced to return to the US.
Survivors say fellow migrants were bound, beaten and even killed just for moving during perilous journey.