Yes you can. The model parameters are still log-odds ratios, but they are estimated differently. Your model with such specifications is basically a nonlinear least squares, where a logit "S" curve is being fit to 0/1 outcomes so as to minimize the squared error. However, the contrasts to usual logistic regression are very well known: this approach puts very little weight on 0/1 outcomes since a proportional difference of 0.95 versus 0.96 is much larger when scaled by its binomial variance. Gaussian families do not assume any mean-variance relationship. That's why this approach is not often used.
If the results given you are proportions, then the burning question is: do you have the denominators for these proportions? e.g. is the 0.43 percent calculated out of $n=100$ or $n=200$ participants and/or does this value differ between the various observations you've obtained? If so, weighting the binomial likelihood gives equivalent inference to fully observed 0/1 counts.
In R, for instance, it will still give you warnings that you have used non-binary outcome variables, but the fitting algorithm does not "break" when inputting data of this format. Other software may prevent such approaches altogether so you will have to create product variables.
However, without such counts in place, other robust error estimation methods should be used. Others' suggestions of quasilikelihood seems like a reasonable choice.