tl;dr: It's legal on any VHF or above frequency where phone modes are permitted. It is not allowed on HF or below, nor where phone modes are not allowed.
What you have described is called "Modulated Continuous Wave" (MCW), as defined in the US amateur radio rules under 47 CFR §97.3(c)(4), and would have the emission designator F2A. (The usual CW used on HF has emission designator A1A.) In areas regulated by the FCC, it is not legal on frequencies below 50.1 MHz, between 144.0 and 144.1 MHz, or between 219 and 220 MHz. It is legal on any other amateur frequency.
MCW is commonly used for automated identification of FM voice repeaters.
It is worth noting that some software capable of transmitting the many HF digital modes via an SSB transceiver can also generate CW in the same fashion. When transmitted this way through an SSB transceiver, it is technically emission designator J2A, which is still within the definition of proper CW. In areas regulated by the FCC, proper CW is legal on any frequency allocated to the amateur service (subject to certain limitations on the 60m band).