Some background

In order to answer the O.P.’s question, a little background must be given regarding how the Catholic Church views marriage.

For any marriage in which one of the parties is (or was) Catholic, generally for the marriage to exist at all, two conditions must be met*:

1. The other party must be at least a baptized Christian (Code of Canon Law [CIC] 1086), and
2. The they are bound to observe the so-called canonical form, which means that the marriage must be witnessed by a person with the necessary faculties: usually a priest or deacon (CIC 1108).

If these conditions are not met, and no other action is taken, the marriage cannot come into existence. The local bishop of the Catholic party, however, can dispense from either or both of these requirements. Such a dispensation is not difficult to obtain, but it must be done in writing. (It is generally written both on the marriage certificate and in the baptismal register.)

In addition, if the non-Catholic party is a baptized Christian, the Catholic party should obtain a license (i.e., permission, not a document like a driver’s license) to marry from his local bishop. This permission is not necessary for the validity of the marriage, but marrying a non-Catholic without it is illicit. (See CIC 1024-1029.)

(There is one exception to all this: if the non-Catholic party is Eastern Orthodox or Oriental Orthodox, after the Catholic party obtains the license from his local bishop, the marriage may be done before a priest of the non-Catholic party’s church. See CIC 1027)

Regarding a Catholic priest giving a blessing at a civil marriage

Returning to the O.P.’s question, therefore, the Catholic Church by default does not recognize a strictly civil marriage involving a Catholic party.

Therefore, unless the couple has the intention of having the religious ceremony before witnesses as canonical form requires (this is common in countries, like Mexico, in which clergy may not function as justices of the peace), or else the couple has obtained a dispensation from canonical form from the bishop, it would be highly improper for a Catholic priest to be present at a civil marriage and give a blessing. (Speaking plainly, he would basically be blessing a fictitious marriage.)

If the non-Catholic party is non-baptized, and the impediment to marriage is not dispensed, then likewise a priest should not give a blessing to the couple.

However, there are remedies that might accomplish what the O.P. is seeking.

If for some reason having the wedding ceremony in a church before properly delegated witnesses is a difficulty, one solution is for the Catholic party to seek a dispensation from canonical form.

If the party is non-baptized, in addition, a dispensation from the impediment due to disparity of cult is necessary. (It can be asked for at the same time.)

Another solution is to have a very simple wedding ceremony in the Church. Note that the wedding need not take place in a Mass. (In fact, it is recommended that marriages with a non-Catholic party not take place during a Mass, to avoid mortifying the non-Catholic party, who may not receive Communion.) In fact, if the non-Catholic is not baptized, the ceremony need not take place in a church. (See CIC 1118.)

If, for whatever reason, the parties go ahead with their (invalid) civil marriage, but later on wish to regularize their situation, they may either opt for a simple convalidation (that is, a simple exchange of consent according to canonical form) or else—especially if having a simple convalidation would cause anguish for one of the spouses—even a radical sanation (which is a convalidation done in secret that does not require a renewal of consent). The necessary dispensations are dealt with at the same time as one seeks the convalidation.

(It should be noted that entering an invalid union is never a good situation to be in, and it should be avoided at all costs. It is much better to encourage a person to seek one of the solutions mentioned above: either dispensation from the requirement of canonical form, or following the canonical form, albeit in as simple a ceremony as the couple desires.)

* Two conditions in addition to the usual conditions for marriage: namely, that the parties consist of one man and one woman, that they are both free to marry, that their decision to marry not due to fear, that they intend to be married lifelong, that they have not absolutely excluded the possibility of having children, and so on. There are also other impediments (see CIC 1083-1094), but these two are the ones most relevant to the O.P.