It would appear so.
There are other references of vedas in Ramayana.
kushiilavau tu dharmaGYau raajaputrau yashasvinau | bhraatarau
svarasampannau dadarsha aashramavaasinau || sa tu medhaavinau
dR^ishhTvaa vedeShu pariniShThitau | vedopabR^ihmaNaarthaaya
taavagraahayata prabhuh || kaavyaM raamaayaNaM kR^itsnaM
siitaayaashcharitaM mahat.h| paulastya vadhamityeva chakaara
The princes, the brothers, Kusha and Lava, were knowledgeable about
Dharma and were ....skilled in the Vedas, and for the sake of
expounding the Vedas, he [Valmiki] composed and made them study the
So, the key is to understand Valmiki composed Ramayana for sake of expounding the Vedas.
First, does the name "rama" occur in the veda ? Yes, it does in Rig Veda. From 10 Mandala, 93 sukta and 14 Rik,
pra tadduHshIme pR^ithavAne vene pra rAme vochamasure maghavatsu | ye
yuktvAya pa.ncha shatAsmayu pathA vishrAvyeshhAm.h ||
nIlakaNTha quotes from the agastya-saMhitA to further support the fact that the rAmAyaNa is drawn from the Vedas:
vedavedaye pare puMsi jAte dasharathAtmaje | vedaH prAchetasAdAsIt.h
sAxAdrAmAyaNAtmanA | tasmAdrAmAyaNaM devi veda eva na saMshayaH ||
nIlakantha also quotes that Valmiki has based the twenty-four thousand verses of the ramayana on the twenty-four syllables of the gayathri mantra. Thus, 1, 1001, 2001 etc. follow the gayathri mantra.He has compiled a list of mantras from the vedas that correspond to the Ramayana. This is known as mantra- rAmAyaNa. One should note that he has written another work based on the correspondence of Rig veda and Srimad Bhagavatam. This is known as mantra-Bhagavata.
five Riks, Rig Veda 10.99.1-5, briefly describe the rAmAyaNa as found in the first few chapters of the vAlmIki rAmAyaNa.
Rig Veda 10.99.1:
kaM nashchitramishhaNyasi chikitvAnpR^ithugmAnaM vAshraM
vAvR^idhadhyai | kattasya dAtu shavaso vyushhTau taxadvajraM
Rig Veda 10.99.2:
sa hi dyutA vidyutA veti sAmapR^ithuM yonimasuratvA sasAda | sa
sanILebhiH prasahAno asya bhrAturna R^ite saptathasya mAyAH |
Now, the translations of the verses are different in various commentaries.
The interpretation of the verses by sayanacharya is based on the assumption that the samhita portion of karma kanda but there is another interpretation by nilakantha saying these are mantra and not to viewed as karma kanda etc.
Clearly, there is no mention of brAhmanas, upanishads, aranyakas in Ramayana. However, the traditional view such as the Sringeri and Kanchi Kamakoti Maths is that Aranyakas, Brahmanas & Upanishads are part of the respective Vedas. It is just that they were all clubbed under Vedas but later in kali yuga, these texts were split into various categories.
Thus, one can only say that Ramayana was composed based on Rig veda around the same time but before the veda was classified/split into aranyakas, brahmanas etc.