No. Only the spouse with the valid social security number qualifies for the stimulus payments. 

One option for you to avoid the population of the Recovery Rebate Credit for your spouse would be to file separately. In that case, his credit would not populate since he is married filing separately without a social security number. 

My spouse actually has a social security number. (It specifies on the card it is not for employment unless authorized by DHS). 

For reference, he is a graduate research assistant (as a F1 student) so he makes money from this position, which is why he has a SSN. 

He may be eligible if he qualifies as a resident alien under the Substantial Presence Test and if his Social Security Card is authorized by DHS. Be very careful though to determine if this is a legitimate authorization. Here are the requirements for him to qualify as a resident alien. If he doesn't qualify as a resident alien, he will not be able to claim the stimulus money. Also if 

1. 31 days during the 2020 year, and
2. 183 days during the 3-year period that includes the current year and the 2 years immediately before that, counting:
   • All the days you were present in the current year, and 

• 1/3 of the days you were present in the first year before the current year, and
• 1/6 of the days you were present in the second year before the current year