Can an apk file be written to any type of NFC tag
Some resources I've found regarding NFC tags seems to indicate this would not be possible:

NFC Chip Types
The highest memory available is only about 7600 bytes. Which is quite a bit more than most apks.

NFC Data Exchange Format
The standardized data format specification that is supported by NFC Devices.


  At a basic level, an NDEF record contains two components: (1) the data and (2) a descriptor of that data which is used to contextualize the data. [ . . . ] NDEF supports a fairly limited set of actions. More complex actions can be implemented with customized software running on the touching device.


So the default actions that may be supported by NFC devices are:


Url
Text
Email
VCard
Phone
SMS
URI


In order to support a non-standard action, you would have to completely implement that in a separate app. Then to transport your apk, it would have to be within the data limitations of all modern NFC tags.