No.
There are a few things worth considering here, though. First of all, the HRE existed for a long time and went through a number of official names. Charles I was crowned Imperator Romanorum in 800, meaning King of the Romans, with "Romans" referring to the people of Rome (the city) rather than all the subjects of the former Roman Empire. The HRE is generally considered to come into existence only in 962 with the coronation of Otto I as Romanorum Imperator Augustus. However, during that time the territories under his control were still referred to as Regnum Francorum (Orientalium), i.e. the Kingdom of the (East) Franks. Back then, when you said "Romans", you were referring to either the citizens of Rome, or the East Roman Empire (a.k.a. the Byzantine Empire) which continued to exist until 1453, and (unlike the HRE) was actually a direct successor of the Roman Empire of antiquity. (The fact that there were two "Roman Emperors" also lead to some diplomatic disputes.)
What we now call the Holy Roman Empire wasn't even called "holy" (sacrum) until 1157 (or 1254, depending on your definition). It's important to note that during this time, political entities were largely derived from titles, rather than vice versa. Today, the President of the United States of America is titled such because there's an entity called the United States of America which bestows this title upon persons of certain political rank. In contrast, the (not-yet-holy) Roman Empire came into being because Charles I styled himself Roman Emperor and made that title hereditary. The POTUS exists (and continues to exist) because of the USA, while the HREmpire existed (and continued to exist) because of the HREmperor.
Likewise, it would be incorrect to consider the HRE a nation-state. National identity in the form it is understood today came into being only around the late 18th century in central Europe. Earlier, identity was based mainly around regional and lingual affiliation. The longer title Sacrum Imperium Romanum Nationis Germanicæ (Heiliges Römisches Reich deutscher Nation / Holy Roman Empire of the German Nation) was first introduced in 1474, and more officially in 1486 and 1512, but the shorter version was much more common. And again, nation back then did not constitute any claim towards political or cultural unity.
In other words, the inhabitants of the HRE weren't all Germans, either. This is very much in line with e.g. inhabitants of the Spanish Netherlands being called Dutch, or Flemish, or whatever, rather than Spanish, and many similar examples.
In the Treaty of Osnabrück (1648), the term "Roman" only occurs in phrases such as these:
You get the idea. The only thing Roman (other than the Roman Catholic Church) was the Roman Empire. In contrast, "German" (and "Germany") are used primarily as regional descriptors, e.g. to differentiate between Germany and Bohemia, both of which were part of the HRE. When the treaty refers to all inhabitants of the HRE, they're just called "subjects" (Vnderthanen), or occasionally "inhabitants" (Jnwohner/Innwohner).
^(Side note: You can find an English translation of the Treaty of Osnabrück at http://avalon.law.yale.edu/17th_century/westphal.asp, but the translation isn't great and can be highly misleading at times.)