Ignoring the similarity of the Kanji used in these two words, could it be possible that 爪 tsume "nail" and 抓る tsuneru "to pinch" are related? This is a question that recently came to mind and I thought I would tackle the idea. In order to do so, I will explore each of their etymological origins separately and attempt to reconcile them together. I will also study the possibility of whether or not both words are of Altaic origin.
Origin of 爪 tsume "nail"
In all of the Japonic variants, there appears to be no other major word used for this term. Instead, we find rather predictable phonological derivations. Most of all Japanese dialects use the form つめ tsume. In Awaji, the variant ちみ chimi is attested in compounds. Shuri Okinawan uses the predictable form ちみ chimi, while Nakijin uses ちみー chimii, and Yonaguni has んみ nmi. These all point to *tsume as the origin.
Origin of 抓る tsuneru "to pinch"
This verb is a little more interesting because we find very little consistency in Japonic variants. In the mainland alone, we find the dialectal variants つめる tsumeru, ひねる hineru, すねずる suneru, つねくる tsunekuru and つねぐる tsuneguru. Perhaps most curious is that we find the Kyoto variant つめる tsumeru, which, on the surface, seems like it could be from 爪 tsume suffixed with the ending -る. However, the variants with -ne(ku)ru- are overwhelmingly more widespread.
In Chiba, the term にじる nijiru is used, which resembles the common Kyushu and Ryukyu variant based on ねずむ nezumu. Most of Kyushu uses the form ねずむ nezumu, with Kagoshima and parts of Miyazaki using the form ねずん nezun. In the Ryukyus, Okinawan uses にじむん nijimun, which is cognate, but it also has the variant にじーん nijiin, which might indicate ねずる nezuru as the origin.
In any case, we clearly note that two major forms have arisen: つねる tsuneru and ねずる nezuru / ねずむ nezumu. It could be possible that one of the two forms is merely an inversion of the other, so that tsuneru became netsuru which evolved into nezuru~nezumu. But if the reverse was true, then the form nezuru would effectively eliminate any possible connection with the word 爪 tsume "nail".
抓る tsuneru and 摘まむ tsumamu "to pinch"
Although it may not be possible to resolve the previous mystery, it is worth noting that we have another word to work with: 摘まむ tsumamu, which also means "to pinch". The -mu ending, while different, seems to be parallel to the common Kyushu form ねずむ nezumu. Given this, and considering that the vowels /e/ and /a/ are known to vary in Japanese, it wouldn't be unthinkable to assume a connection with 爪 tsume. In fact, 摘まむ tsumamu is actually a derivative of the basic form 摘む tsumu, meaning "to pluck; to pick". This gives us the certain root tsum-.
Now, if we assumed that 摘む tsumu historically gave way to derived verb 摘むる tsumuru, which would have 摘める as its modern reflex, then it would be possible to establish a connection. Such a verb would have つめ tsume for its nominal form, which would theoretically mean "that which picks/plucks/pinches". If we assume that 抓る tsuneru is, in fact, that reflex which changed slightly over time, then we could state that they're undoubtedly related. But given the widespread use of tsuneru and nezuru, I hesitate to conclude this as a certainty, as the change would have to predate Old Japanese.
Nonetheless, a connection between 摘む tsumu and 爪 tsume seems likely.
The Altaic connection
Although the Altaic proposal is generally regarded with skepticism, it is worth noting the following Turkic words for "nail":
Bashkir: тырнаҡ (tïrnaq)
Crimean Tatar: tırnaq
Kazakh: тырнақ (tırnaq)
Kyrgyz: тырмак (tırmaq)
Tatar: тырнақ (tırnaq)
Most interesting is that at least one variant has -m- where the rest have -n-. Wikipedia also cites Old Turkic as having the forms tırŋak and tarmak. Semantically, these words also cover the meanings of "claw" and "hoof", which is similar to the Japanese term 爪 tsume.
If we were to assume a connection, then Japanese would expectantly have lost the coda-final consonants, giving *tïna~tïma. Considering that some authors have hypothesized that Japanese once nominalized words by affixing them with -i, then we would presumably obtain *tïnai~tïmai and later *tïne~tïme as a result of vowel coalescence. Assuming the pseudo vowel /ï/ became /u/ at some point, then we would end up with the possible modern variation of /tune~tume/ tsune~tsume, with /tuma/ tsuma- and /tuna/ tsuna- occurring more rarely in compounds.
Could this indicate a relationship or an early loan? Who knows. The book Indo-European and its closest relatives. 2. Lexicon , by Joseph Harold Greenberg, compares other Altaic languages and suggests a relationship.
Food for thought.