I think there are some coincidences, but for the most part I don't think you can just use a counter suffix as its own noun. Unless you're explicitly using it as the topic of a sentence to describe what the counter means.
匹【ひき】 (counter for animals) could not be used as a noun by itself...unless your sentence is 「『匹』とは小さな動物を数える助数詞です。」("Hiki" is a counter for counting small animals.
I previously said that there are some coincidences. For example,
一本【いっぽん】 means 1 cylindrical object. Now
本 by itself is a word -- "book". But you would never use it by itself in the context of a cylindrical object; only as a book.
〜頭【とう】 is used for counting "heads of animals" like cows, etc. (see posts HERE and HERE for discussions about which animals can be counted this way).
頭 by itself is
あたま (or less frequently,
こうべ) and means "head". But you would never use
頭 by itself in the context of "heads of animals".