Seems odd to say this, but the last (and only) time I looked at “ring”–as in ring or ring up someone on the phone, or ring off, was more than ten years ago, in 2011. I tagged it as “On the radar” then, and I’d have to say that’s still the case, as the first time I’ve encountered a U.S. use since then was last week, when I was listening to the Gimlet podcast “Heavyweight” (highly recommended), when this promo for another Gimlet show, “Every Little Thing,” came on:
It’s the phone number. Maybe “ring” is starting to make inroads, or maybe it’s just that CALL-ELT wasn’t available.
notoneoffbritishisms.com 
For me, this is an example of the beginning of the deterioration of the English language. The word ‘telephone’ does have a structure, traceable to previous developments of our language, but when we needed a verb to describe what was done with a telephone, there was an immediate collapse into ‘telephone'(verb), ‘call’ and ‘ring’. The forming of a verb from a noun has grown considerably and must please all advocates of Newspeak.
English nouns have always become verbs. It’s a basic feature of the language. There’s an estimate that something like 20% of English verbs began as nouns. (You’ve used an obvious one yourself — “form”.)
https://www.thoughtco.com/what-is-verbing-1691035
This raises one of the common objections, such as the claim that the language is constantly developing and growing. What I was writing about was the use of a noun as a verb because of illiteracy, ignorance or laziness. More examples are knife, bin, rubbish. This is is verb reduction.
Soon, there will be few adjectives and verbs left.
I suspect ring up in its telephone usage has not taken off due the use of “ring up” being used for the process of checking out at grocery and other stores (I have no idea if that is a regionalism, an older usage, or widespread in the US, but that is the one I grew up with). I run into the call “ring up” in a fair number of fiction books (usually involving British people or Anglophile characters), but haven’t really heard it used otherwise.
I don’t think there is an equivalent in British English to the US ‘ring up’ (as in going through a checkout process).
Are they called ringtones in America?
According to the OED, “ringtone” originally (as of 1921) meant “The sound produced in a caller’s telephone to indicate that a connection with the number dialled has been made and the called phone is ringing.” Then (circa 1984) it took on the meaning of “A sound made by a telephone when there is an incoming call awaiting an answer; esp. one made by a mobile phone, typically a snatch of music or other sound chosen from a preselected list, or available to download or buy from the internet.” That sense is indeed used in the U.S.
I remember the publicity in the sixties when Post Office Telephones introduced the Trimphone with its warbling sound. It was said that birds started imitating the sound.
What do Americans say when the phone indicates that there is an incoming call? Do you not say that the phone is ‘ringing’?
Yes, in the U.S., we say that the phone rings, but we usually don’t say that we ring someone; we say that we call them or phone them.
Yes, we do.
With the numerous means of contacting people these days by voice or video (eg. WhatsApp, Messenger), “call” is universally acceptable both as a verb and noun. “Phone” and “ring” are specifically restricted to use of a telephone, either land-line or mobile (cellphone). Although “ring” is used as a verb in British English, I think it is more common as a noun, as in “I’ll give you a ring.”
How about “Give me a bell on the blower”? 🙂
Perhaps the ghastly new one imported from the USA will spread to telephone calls: reaching out.