I don't see him having a flush here very often. He would usually make a move on the flop with a flush draw (considering most likely combinations out there contain a flush + straight draw or pair, or nut flush draw) and he should definitely want to get more money in the pot on the turn if he hit (as well as protect his hand if it isn't the nuts).
His range looks something like missed draws (JT, 67, w or w/o a heart) and hands like 9x trying to get to a cheapish showdown. He could also have a hand like
T or Kx but I think both these are fairly unlikely (oop float) and can basically be disregarded as they 'cancel' each other out.
I would call, but it may be a mistake. Lately however I've been seeing a lot of people making weak-ass bets on the river as bluffs (because I'm a CS when it comes to the river) so I think calling here would be correct. If it is wrong, it isn't wrong by much. Basically 90% of these situations and hands can be summed up by saying "it doesn't really matter too much what you do".
Oh I totally disregarded the quick turn call. That's generally an indication of a drawing hand - more of a reason to call. If he had a big hand he would at least consider raising and a mediocre hand should consider folding.