God's existence and divinity can be demonstrated.

Vatican I

The First Vatican Council defined the dogma of the knowability of God from natural reason in Dei Filius II canon 1 on Revelation:

1. If any one shall say that the one true God, our Creator and Lord, cannot be certainly known by the natural light of human reason through created things: let him be anathema.
   
   ^{[Cdl. Manning's transl. of:] Si quis dixerit, Deum unum et verum, Creatorem et Dominum nostrum, per ea, quæ facta sunt, naturali rationis humanæ lumine certo cognosci non posse: anathema sit}

The beginning of ch. 2 on Revelation says:

The same holy Mother Church holds and teaches that God, the beginning and end of all things, may be certainly known by the natural light of human reason, by means of created things; "for the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made," [Rom. 1:20]

^{[Cdl. Manning's transl. of:] Eadem sancta mater Ecclesia tenet et docet, Deum, rerum omnium principium et finem, naturali humanæ rationis lumine e rebus creatis certo cognosci posse; invisibilia enim ipsius, a creatura mundi, per ea quæ facta sunt, intellecta, conspiciuntur}

"the mind of the Council was in favor of [God's] demonstrability"

Sacræ Theologiæ Summa IIA: One & Triune God, "Thesis 2: The existence of God can be demonstrated scientifically and intellectually," p. 29:

Certainly Vatican I deliberately omitted the word “demonstration” in its definition. [CL 7.132] That would seem to be easily explained, as Lennerz [De Deo uno, 64] notes, from the intention of the Council of defining a certain principle against strict traditionalism [a kind of fideism] by which also the milder form would necessarily be included. But that the mind of the Council was in favor of demonstrability is sufficiently clear from the fact that it was concerned about “the widely circulated error that the existence of God cannot be proved with any certain arguments and therefore cannot be known by the power of reason,” and “the arguments that have always been highly regarded are not of such a nature as to be convincing.” [CL 7.79] Cardinal Gasser himself conceded that “in some measure, to know for certain and to demonstrate” are “one and the same thing.” [CL 7.132] Therefore, according to the mind of the Council demonstrability is closely connected with certain knowledge, and Pius XI rightly said that the oath prescribed by Pius X [quoted below] excellently interpreted Vatican I.

Magisterial documents explicitly using the word "demonstrate"

Leo XIII, encyclical Æterni Patris §§4,5 (1879):

certain truths which were either divinely proposed for belief, or were bound by the closest chains to the doctrine of faith, were discovered by pagan sages with nothing but their natural reason to guide them, were demonstrated and proved by becoming arguments. […] In the first place, then, this great and noble fruit is gathered from human reason, that it demonstrates that God is

^{[Latin:] quaedam vera, quae vel divinitus ad credendum proponuntur, vel cum doctrina fidei arctis quibusdam vinculis colligantur, ipsi ethnicorum sapientes, naturali tantum ratione praelucente, cognoverint, aptisque argumentis demonstraverint ac vindicaverint […] primo loco magnus hic et praeclarus ex humana ratione fructus capitur, quod illa Deum esse demonstret}

St. Pius X, encyclical Iucunda sane §15 (1904):

Men even go so far as to impugn the arguments demonstrating the existence of God, denying with unparalleled audacity and against the first principles of reason the invincible force of the proof which from the effects ascends to their cause, that is God, and to the notion of His infinite attributes.

^{[Latin:] Impetuntur ipsa argumenta, quibus Deum esse demonstratur, atque incredibili temeritate, contra prima rationis indicia, repudiatur invicta illa argumentandi vis, qua ex effectibus causa colligitur, id est Deus eiusque attributa, nullis circumscripta limitibus.}

Pius XI said that St. Pius X's Oath Against Modernism (Sacrorum antistitum, 1910) correctly interprets Vatican I's Dei Filius. Sacrorum antistitum (DZ 2145) says:

I profess that God, the beginning and end of all things, can be certainly known and thus can also be demonstrated by the natural light of reason "by the things that are made" [cf. Rom. 1:20], that is, by the visible works of creation, as the cause by the effects.

^{[Latin: DZ 3538:] Deum, rerum omnium principium et finem naturali rationis lumine 'per ea quae facta sunt' (cf. Rom 1, 20) , hoc est, per visibilia creationis opera, tamquam causam per effectus, certo cognosci, adeoque demonstrari etiam posse, profiteor.}

^{source for these quotes: Sacræ Theologiæ Summa IIA: One & Triune God, "Thesis 2: The existence of God can be demonstrated scientifically and intellectually," pp. 27-29.}