Do aircraft manufacturers have the authority to either approve or disapprove the maintenance procedures of its customers?
For N-Registered Planes Under FAA regulations... (this may differ elsewhere in the world)

The short answer is No (although some service bulletins may be legally required), the manufacturer can not legally enforce anything per say, however here's how it plays out. 

Most makers are at least in some capacity always testing their aircraft and aircraft procedures (at least the models in service). This may be as simple as inspecting the ones out there and looking at used parts for fatigue and what not. This in turn can lead to the company issuing a Service Bulletin which is basically a document that says: "hey here's something we found, you should look at it". This can apply to a any part of the aircraft as well as inspection and maintenance procedures. However service bulletins are not legally required to be complied with when issued by the company (although many may choose to comply). 

On the other hand the Service Bulletins may trigger the FAA to issue an Airworthiness Directive which is a legally enforceable document. In this case the owner/operator of the aircraft is legally required to comply and alter their procedures.  

In this case of the accident in question I think this is the AD that was issued as a result. You can find a full list of MD-10 AD's Here.

On to the question of why an airline may not listen. This is most likely a cost related issue. If the airlines insurance does not change and the cost to inspect or possibly replace a part that they don't legally need to replace is simply to high, they wont do so until its legally required. Again since every thing in aviation has two sides, manufactures may have the courtesy to replace the part at no cost to the owner. For example, when Porsche dropped the FAA Type Certification for the 3.2 motor in the Mooney M20PFM leaving all the owners somewhat high and dry, Mooney was kind enough to have the 40 odd aircraft updated with Continental motors (although many were destroyed in a hurricane while under going the change). The crux of it also comes down to the fact that only the government and its various organizations (here in US) can legally impose anything on an individual or company. So while it may be a good idea to heed the advice of the maker you may not be required to do so.  

--Edit--

To answer the comment below which is somewhat relevant to the original question. My understanding is that yes the FAR's may require you to refer to the manufacturer procedure but at the same time you only must refer to the procedures required (associated with the current certification of the airframe) for your airframe an SB with out an accompanying AD is a manufacturer procedure but is not legally required. So you may refer to it but you may not have to comply with it. Ill dig into the regulations and see if I can find the letter of the law on it.