I completed a four-year Aspirant course and successfully defended my Aspirant thesis
Apparently you don't have a Kandidat Nauk degree (otherwise you should have wrote about it), so by no means it is equivalent to PhD. Aspirantura without a Kandidat Nauk is not valued even in Russia.
Traditionally, Kandidat Nauk dissertation and degree was almost the only outcome of study in aspirantura. However, completion of aspirantura by itself did not automatically result in dissertation defence and awarding a degree. There were and there are a lot of people who do not submit and defend a dissertation while in aspirantura, so they would complete aspirantura without getting a degree. In this case they usually would not even get any diploma. (Some of them would defend a dissertation later, some would outright drop.) There were no final exams in aspirantura, and often no advanced curriculum; and in this sense, aspirantura was more kind of a bureaucratic device that allowed to get funding, give 'legal status' to aspirantura students, not an education institute itself. The main aim of aspirantura was not to give education to students, but to allow them to do research.
However, in recent years, Russian government (Ministry of Education, I suppose) decided to fit aspirantura into the main framework of Russian Federal Law on Education. In particular, AFAIK, aspirantura students are now awarded some diploma on successful completion of aspirantura. It is not tied to Kandidat Nauk dissertation, and has much less requirements, and mainly signifies that some person has indeed studied in aspirantura, and completed some aspirantura courses, and maybe some simple final exam. Even some thesis may be required, but I guess that requirements for such thesis are much lower than for a candidate dissertation.
This aspirantura diploma is not really valued in Russia. (On contrary, it can be a somewhat negative signal, because successfull aspirantura students will defend a Kandidat dissertation and thus will have Kandidat Nauk status, so if somebody has only aspirantura diploma and not a Kandidat Nauk one, this means that they tried and failed). The only use case of aspirantura diploma that I have heard of is when one has to prove that he had received a specific number of years of education, usually when they apply to some position in another country. For example, I've heard it used when one applies to European medical PhD programs. They require that applicants have, e.g., 8 years of medical education (the figures are approximate), while Russian medical universities only provide e.g. 6 years, so additional 3-4 aspirantura years make for the difference.
So if you only have an aspirantura diploma (and defended some thesis required for this diploma), then it is not a Kandidat Nauk degree, and thus absolutely not a PhD equivalent.
Disclaimer: I do not have any direct relation to Russian science in recent years, so my knowledge is based on rumors and information from my friends, many of whom are still working in scientific institutes, and some study in aspirantura, and so the information above may be not exactly correct.
As soon as you defended the Dissertation, the Dissertation Panel awards you the Kandidat Nauk title. It shall be later vetted by VAK (the Higher Attestation Committee), which actually issues the Diploma to confirm your Kandidat Nauk title, but technically speaking you bear the title from the moment the successful defence is announced.
Anyway, as soon as you have a Kandidate Nauk diploma, you can apply to PhD-level jobs in Russia and abroad (including USA). As with any PhD and any graduate degree in general, some people pay a lot of attention to which institution has awarded this degree (e.g. some would seriously consider only applicants with PhD from top-10 universities in the world), and some people/organisations do not have any preferences and are happy to accept any candidate with PhD from a University of Obscure Neverland. Some USA universities may require your Diploma to be notarised and officially recognised, some may accept it with translation you made yourself. You need to check with each University what the rules and expectations are.